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How Bell’s Theorem Proved ‘Spooky Action at a Distance’ Is Real

July 20, 2021

Samuel Velasco/Quanta Magazine

Introduction

We take for granted that an event in one part of the world cannot instantly affect what happens far away. This principle, which physicists call locality, was long regarded as a bedrock assumption about the laws of physics. So when Albert Einstein and two colleagues showed in 1935 that quantum mechanics permits “spooky action at a distance,” as Einstein put it, this feature of the theory seemed highly suspect. Physicists wondered whether quantum mechanics was missing something.

Then in 1964, with the stroke of a pen, the Northern Irish physicist John Stewart Bell demoted locality from a cherished principle to a testable hypothesis. Bell proved that quantum mechanics predicted stronger statistical correlations in the outcomes of certain far-apart measurements than any local theory possibly could. In the years since, experiments have vindicated quantum mechanics again and again.

Bell’s theorem upended one of our most deeply held intuitions about physics, and prompted physicists to explore how quantum mechanics might enable tasks unimaginable in a classical world. “The quantum revolution that’s happening now, and all these quantum technologies — that’s 100% thanks to Bell’s theorem,” says Krister Shalm , a quantum physicist at the National Institute of Standards and Technology.

Here’s how Bell’s theorem showed that “spooky action at a distance” is real.

Ups and Downs

The “spooky action” that bothered Einstein involves a quantum phenomenon known as entanglement, in which two particles that we would normally think of as distinct entities lose their independence. Famously, in quantum mechanics a particle’s location, polarization and other properties can be indefinite until the moment they are measured. Yet measuring the properties of entangled particles yields results that are strongly correlated, even when the particles are far apart and measured nearly simultaneously. The unpredictable outcome of one measurement appears to instantly affect the outcome of the other, regardless of the distance between them — a gross violation of locality.

To understand entanglement more precisely, consider a property of electrons and most other quantum particles called spin. Particles with spin behave somewhat like tiny magnets. When, for instance, an electron passes through a magnetic field created by a pair of north and south magnetic poles, it gets deflected by a fixed amount toward one pole or the other. This shows that the electron’s spin is a quantity that can have only one of two values: “up” for an electron deflected toward the north pole, and “down” for an electron deflected toward the south pole.

Imagine an electron passing through a region with the north pole directly above it and the south pole directly below. Measuring its deflection will reveal whether the electron’s spin is “up” or “down” along the vertical axis. Now rotate the axis between the magnet poles away from vertical, and measure deflection along this new axis. Again, the electron will always deflect by the same amount toward one of the poles. You’ll always measure a binary spin value — either up or down — along any axis.

It turns out it’s not possible to build any detector that can measure a particle’s spin along multiple axes at the same time. Quantum theory asserts that this property of spin detectors is actually a property of spin itself: If an electron has a definite spin along one axis, its spin along any other axis is undefined.

Local Hidden Variables

Armed with this understanding of spin, we can devise a thought experiment that we can use to prove Bell’s theorem. Consider a specific example of an entangled state: a pair of electrons whose total spin is zero, meaning measurements of their spins along any given axis will always yield opposite results. What’s remarkable about this entangled state is that, although the total spin has this definite value along all axes, each electron’s individual spin is indefinite.

Suppose these entangled electrons are separated and transported to distant laboratories, and that teams of scientists in these labs can rotate the magnets of their respective detectors any way they like when performing spin measurements.

When both teams measure along the same axis, they obtain opposite results 100% of the time. But is this evidence of nonlocality? Not necessarily.

Alternatively, Einstein proposed, each pair of electrons could come with an associated set of “hidden variables” specifying the particles’ spins along all axes simultaneously. These hidden variables are absent from the quantum description of the entangled state, but quantum mechanics may not be telling the whole story.

Hidden variable theories can explain why same-axis measurements always yield opposite results without any violation of locality: A measurement of one electron doesn’t affect the other but merely reveals the preexisting value of a hidden variable.

Bell proved that you could rule out local hidden variable theories, and indeed rule out locality altogether, by measuring entangled particles’ spins along different axes.

Suppose, for starters, that one team of scientists happens to rotate its detector relative to the other lab’s by 180 degrees. This is equivalent to swapping its north and south poles, so an “up” result for one electron would never be accompanied by a “down” result for the other. The scientists could also choose to rotate it an in-between amount — 60 degrees, say. Depending on the relative orientation of the magnets in the two labs, the probability of opposite results can range anywhere between 0% and 100%.

Without specifying any particular orientations, suppose that the two teams agree on a set of three possible measurement axes, which we can label A, B and C. For every electron pair, each lab measures the spin of one of the electrons along one of these three axes chosen at random.

Let’s now assume the world is described by a local hidden variable theory, rather than quantum mechanics. In that case, each electron has its own spin value in each of the three directions. That leads to eight possible sets of values for the hidden variables, which we can label in the following way:

The set of spin values labeled 5, for instance, dictates that the result of a measurement along axis A in the first lab will be “up,” while measurements along axes B and C will be “down”; the second electron’s spin values will be opposite.

For any electron pair possessing spin values labeled 1 or 8, measurements in the two labs will always yield opposite results, regardless of which axes the scientists choose to measure along. The other six sets of spin values all yield opposite results in 33% of different-axis measurements. (For instance, for the spin values labeled 5, the labs will obtain opposite results when one measures along axis B while the other measures along C; this represents one-third of the possible choices.)

Thus the labs will obtain opposite results when measuring along different axes at least 33% of the time; equivalently, they will obtain the same result at most 67% of the time. This result — an upper bound on the correlations allowed by local hidden variable theories — is the inequality at the heart of Bell’s theorem.

Above the Bound

Now, what about quantum mechanics? We’re interested in the probability of both labs obtaining the same result when measuring the electrons’ spins along different axes. The equations of quantum theory provide a formula for this probability as a function of the angles between the measurement axes.

According to the formula, when the three axes are all as far apart as possible — that is, all 120 degrees apart, as in the Mercedes logo — both labs will obtain the same result 75% of the time. This exceeds Bell’s upper bound of 67%.

That’s the essence of Bell’s theorem: If locality holds and a measurement of one particle cannot instantly affect the outcome of another measurement far away, then the results in a certain experimental setup can be no more than 67% correlated. If, on the other hand, the fates of entangled particles are inextricably linked even across vast distances, as in quantum mechanics, the results of certain measurements will exhibit stronger correlations.

Since the 1970s, physicists have made increasingly precise experimental tests of Bell’s theorem. Each one has confirmed the strong correlations of quantum mechanics. In the past five years, various loopholes have been closed. Physicists continue to grapple with the implications of Bell’s theorem, but the standard takeaway is that locality — that long-held assumption about physical law — is not a feature of our world.

Editor’s note: The author is currently a postdoctoral researcher at JILA in Boulder, Colorado.

Clarification August 19, 2021: This article was revised to remove the impression that the standard interpretation of Bell’s theorem is universally accepted among physicists.

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• Published: 31 August 2021

How the Bell tests changed quantum physics

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Nature Reviews Physics volume  3 ,  pages 674–676 ( 2021 ) Cite this article

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More than 40 years ago the first Bell tests translated a purely philosophical conundrum to a physical experiment. In doing so, they changed our understanding of quantum mechanics and contributed to the development of quantum technologies.

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Bell’s Inequalities Experiments

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The tests of Bell’s inequalities were the first and most important experiments in quantum foundations. This chapter presents their origins, historical background, conceptual formulation, and requirements. In the 1960s, thanks to a paper by John Stuart Bell, it became possible to turn thought experiments conceived in the early prewar debates among the founders of quantum mechanics into actual experiments. Bell proposed a theorem that allowed physicists to compare the predictions of QM against those of alternative local hidden-variable theories (LHVT) experimentally. John Clauser, Michael Horne, Abner Shimony, and Richard Holt took up the challenge. They proposed a simple experimental setup comprising a source of entangled photons, analyzers, and detectors connected to a coincidence circuit that was the basis of the first foundational experiments.

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For an account of recent Bell tests see David Kaiser, “Tackling Loopholes in Experimental Tests of Bell’s Inequality,” in The Oxford Handbook of the History of Interpretations of Quantum Physics , eds. Olival Freire Jr. et al. (Oxford University Press, 2022). For a first-hand account by one of the leaders of the field that shows how lively the prospects are for new experiments, see Anton Zeilinger, Dance of the Photons: From Einstein to Quantum (New York: Farrar, Straus and Giroux, 2010). Part of the reason for the continued investment in tests of Bell’s inequalities with ever more sophisticated instrumentation is the role it plays in promising technologies such as quantum computers and cryptography.

For details of how the EPRB thought experiment was brought to the laboratory, see Olival Freire Jr., “Philosophy Enters the Optics Laboratory: Bell’s Theorem and Its First Experimental Tests (1965–1982),” SHPMP 37, no. 4 (2006): 577–616. See also, for more conceptual presentations, M. A. B. Whitaker, “Theory and Experiment in the Foundations of Quantum Theory,” Progress in Quantum Electronics 24, no. 1 (2000): 1–106; Francisco J. Duarte, Fundamentals of Quantum Entanglement (Bristol, UK: IOP Publishing, 2019).

John Clauser, Michael Horne, Abner Shimony, and Richard Holt, “Proposed Experiment to Test Separable Hidden-Variable Theories,” PRL 23, no. 15 (1969): 880–884. See also John Clauser and Abner Shimony, “Bell’s Theorem. Experimental Tests and Implications,” Reports on Progress in Physics 41, no. 12 (1978): 1881–1927.

Michael Horne, Abner Shimony, and Anton Zeilinger, “Down-Conversion Photon Pairs: A New Chapter in the History of Quantum Mechanical Entanglement,” in Quantum Coherence: Proceedings, International Conference on Fundamental Aspects of Quantum Theory 1989, ed. Jeeva S. Anandan (Singapore: World Scientific, 1990), 356–72, p. 362.

C. S. Wu and I. Shaknov, “The Angular Correlation of Scattered Annihilation Radiation,” PR 77, no. 1 (1950): 136; Carl Kocher and Eugene Commins, “Polarization Correlation of Photons Emitted in an Atomic Cascade,” PRL 18, no. 15 (1967): 575–7. Although the paper CHSH was the one which led to the first Bell’s tests, it was not the first to point out that Bell’s theorem could be tested in a laboratory. In 1968, the Berkeley physicist Henry Stapp showed in a preprint that Bell’s theorem could be tested in proton-scattering, the topic of this PhD dissertation. The test would be likewise a variation of experiments that had already been performed to measure angles other than 0° and 90° for the orientation between the analyzers. See David Kaiser, How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival (New York: W. W. Norton, 2011) , pp. 55–56. Stapp’s suggestion was actually materialized in an experiment in 1976, but this will not be discussed here.

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Does Bell's Inequality rule out local theories of quantum mechanics?

In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics.  This so-called "EPR Paradox" has led to much subsequent, and still ongoing, research.  This article is an introduction to EPR, Bell's Inequality, and the real experiments that have attempted to address the interesting issues raised by this discussion.

One of the principal features of quantum mechanics is that not all the classical physical observables of a system can be simultaneously well defined with unlimited precision, even in principle.  Instead, there may be several sets of observables that give qualitatively different, but nonetheless complete (maximal possible), descriptions of a quantum mechanical system.  These sets are sets of "good quantum numbers," and are also known as "maximal sets of commuting observables."  Observables from different sets are "noncommuting observables".

A well known example is position and momentum.  We can put a subatomic particle into a state of well-defined momentum, but then the value of its position is completely ill defined.  This is not a matter of an inability to measure position to some accuracy; rather, it's an intrinsic property of the particle, no matter how good our measuring apparatus is.  Conversely, you can put a particle in a definite position, but then the value of its momentum is completely ill defined.  We can also create states of intermediate "knowledge" of both observables: if you confine the particle to some arbitrarily large region of space, you can define the value of its momentum more and more precisely.  But the particle can never have well-defined values of both position and momentum at the same time.  When physicists speak of how much "knowledge" they have of two noncommuting observables in quantum mechanics, they don't mean that those observables both have well-defined values that are not quite known; rather, they mean the two observables do not have completely well-defined values.

(Technically speaking, the situation is a little more complicated.  Even for observables that don't commute, it is sometimes possible for both to have well-defined values.  Such subtleties are very important to those who examine the derivation of Bell's Inequality in great detail to find hidden assumptions.  For the purposes of this short article, we'll overlook these finer points.)

Position and momentum are continuous observables.  But the same situation can arise for discrete observables, such as spin.  The quantum mechanical spin of a particle along each of the three space axes is a set of mutually noncommuting observables.  We can only know the spin along one axis at a time.  A proton with spin "up" along the x -axis has an undefined spin value along the y and z axes.  We cannot simultaneously measure the x and y spin projections of a proton.  EPR sought to demonstrate that this phenomenon could be exploited to construct an experiment that would demonstrate a paradox that they believed was inherent in the quantum mechanical description of the world.

They imagined two physical systems that are allowed to interact initially so that they will subsequently be defined by a single quantum mechanical state.  (For simplicity, imagine a simple physical realization of this idea—a neutral pion at rest in your lab, which decays into a pair of back-to-back photons.  The pair of photons is described by a single two-particle wave function.)  Once separated, the two systems (read: photons) are still described by the same wave function, and a measurement of one observable of the first system will determine the measurement of the corresponding observable of the second system.  (Example: the neutral pion is a scalar particle—it has zero angular momentum.  So the two photons must speed off in opposite directions with opposite spin.  If photon 1 is found to have spin up along the x -axis, then photon 2 must have spin down along the x -axis, since the total angular momentum of the final-state, two-photon, system must be the same as the angular momentum of the initial state, a single neutral pion.  We know the spin of photon 2 even without measuring it.)  Likewise, the measurement of another observable of the first system will determine the measurement of the corresponding observable of the second system, even though the systems are no longer physically linked in the traditional sense of local coupling.

(By "local" is meant that influences between the particles must travel in such a way that they pass through space continuously; i.e. the simultaneous disappearance of some quantity in one place cannot be balanced by its appearance somewhere else if that quantity didn't travel, in some sense, across the space in between.  In particular, this influence cannot travel faster than light, to preserve relativity theory.)

QM creates the puzzling situation in which the first measurement of one system should "poison" the first measurement of the other system, no matter what the distance between them.  (In one commonly studied interpretation, the mechanism by which this proceeds is "instantaneous collapse of the wave function".  But the rules of QM do not require this interpretation, and several other perfectly valid interpretations exist.)  One could imagine the two measurements were so far apart in space that special relativity would prohibit any influence of one measurement over the other.  For example, after the neutral-pion decay, we can wait until the two photons are light years apart, and then "simultaneously" measure the x -spin of the photons.  QM suggests that if say the measurement of photon 1's x -spin happens first, then this measurement must instantaneously force photon 2 into a state of well-defined x -spin, even though it is light years away from photon 1.

How do we reconcile the fact that photon 2 "knows" that the x -spin of photon 1 has been measured, even though they are separated by light years of space and far too little time has passed for information to have travelled to it according to the rules of special relativity?  There are basically two choices.  We can accept the postulates of QM as a fact of life, in spite of its seemingly uncomfortable coexistence with special relativity, or we can postulate that QM is not complete: that there was more information available for the description of the two-particle system at the time it was created, but that we didn't know that information, perhaps because it cannot be known in principle, or perhaps because QM is currently incomplete.

So, EPR postulated that the existence of such "hidden variables", some currently unknown properties, of the systems should account for the discrepancy.  Their claim was that QM theory is incomplete: it does not completely describe physical reality.  System II knows all about System I long before the scientist measures any of the observables, thereby supposedly consigning the other noncommuting observables to obscurity.  Furthermore, they claimed that the hidden variables would be local , so that no instantaneous action at a distance would be necessary.  Niels Bohr, one of the founders of QM, held the opposite view that there were no hidden variables.  (His interpretation is known as the "Copenhagen Interpretation" of QM.)

In 1964 John Bell proposed a mechanism to test for the existence of these hidden variables, and he developed his famous inequality as the basis for such a test.  He showed that if the inequality were ever not satisfied, then it would be impossible to have a local hidden variable theory that accounted for the spin experiment.

Using the example of two photons configured in the singlet state, consider this: in the hidden variable theory, after separation, each photon will have spin values for each of the three axes of space, and each spin will have one of two values; call them "+" and "−".  Call the axes x , y , z , and call the spin on the x -axis x + if it is "+" on that axis; otherwise call it x− .  Use similar definitions for the other two axes.

Now perform the experiment.  Measure the spin on one axis of one photon and the spin in another axis of the other photon.  If EPR were correct, each photon will simultaneously have properties for spin in each of axes x , y and z .

Next, look at the statistics.  Perform the measurements with a number of sets of photons.  Use the symbol N(x+, y−) to designate the words "the number of photons with x+ and y− ".  Similarly for N(x+, y+) , N(y−, z+) , etc.  Also use the designation N(x+, y−, z+) to mean "the number of photons with x+ , y− and z+ ", and so on.  It's easy to demonstrate that for a set of photons

because the z+ and z− exhaust all possibilities.  We can make this claim if these measurements are connected to some real properties of the photons.

Let n[x+, y+] be the designation for "the number of measurements of pairs of photons in which the first photon measured x+ , and the second photon measured y+ ".  Use a similar designation for the other possible results.  This is necessary because this is all that is possible to measure.  We can't measure both x and y for the same photon.  Bell demonstrated that in an actual experiment, if (1) is true (indicating real properties), then the following must be true:

Additional inequality relations can be written by just making the appropriate permutations of the letters x , y and z and the two signs.  This is Bell's Inequality, and it is proved to be true if there are real (perhaps hidden) variables to account for the measurements.

At the time Bell's result first became known, the experimental record was reviewed to see if any known results provided evidence against locality.  None did.  Thus an effort began to develop tests of Bell's Inequality.  A series of experiments was conducted by Aspect ending with one in which polarizer angles were changed while the photons were in flight.  This was widely regarded at the time as being a reasonably conclusive experiment that confirmed the predictions of QM.

Three years later, Franson published a paper showing that the timing constraints in this experiment were not adequate to confirm that locality was violated.  Aspect measured the time delays between detections of photon pairs.  The critical time delay is that between when a polarizer angle is changed and when this affects the statistics of detecting photon pairs.  Aspect estimated this time based on the speed of a photon and the distance between the polarizers and the detectors.  Quantum mechanics does not allow making assumptions about where a particle is between detections.  We cannot know when a particle traverses a polarizer unless we detect the particle at the polarizer.

Experimental tests of Bell's Inequality are ongoing, but none has yet fully addressed the issue raised by Franson.  In addition there is an issue of detector efficiency.  By postulating new laws of physics, one can get the expected correlations without any nonlocal effects unless the detectors are close to 90% efficient.  The importance of these issues is a matter of judgment.

The subject is alive theoretically as well.  Eberhard and later Fine uncovered further subtleties in Bell's argument.  Some physicists argue that it may be possible to construct a local theory that does not respect certain assumptions in the derivation of Bell's Inequality.  The subject is not yet closed, and may yet provide more interesting insights into the subtleties of quantum mechanics.

Bell's theorem

Curator: Nino Zanghi

Eugene M. Izhikevich

Travis Norsen

Sheldon Goldstein

Daniel Victor Tausk

Riccardo Guida

Carlo Maria Becchi

Nathan Argaman

Sheldon Goldstein , Mathematics Department, Rutgers University, NJ

Travis Norsen , Smith College, Northampton, MA, USA

Daniel Victor Tausk , Mathematics Department, University of São Paulo, Brazil

Dr. Nino Zanghi , Physics Department, University of Genoa, Italy

Bell's theorem asserts that if certain predictions of quantum theory are correct then our world is non-local. "Non-local" here means that there exist interactions between events that are too far apart in space and too close together in time for the events to be connected even by signals moving at the speed of light. This theorem was proved in 1964 by John Stewart Bell and has been in recent decades the subject of extensive analysis, discussion, and development by both physicists and philosophers of science. The relevant predictions of quantum theory were first convincingly confirmed by the experiment of Aspect et al. in 1982; they have been even more convincingly reconfirmed many times since. In light of Bell's theorem, the experiments thus establish that our world is non-local. This conclusion is very surprising, since non-locality is normally taken to be prohibited by the theory of relativity.

Historical background

John Bell's interest in non-locality was triggered by his analysis of the problem of hidden variables in quantum theory and in particular by his learning about the de Broglie–Bohm 1 "pilot-wave" theory (aka "Bohmian mechanics" 2 ). Bell wrote that David "Bohm's 1952 papers on quantum mechanics were for me a revelation. The elimination of indeterminism was very striking. But more important, it seemed to me, was the elimination of any need for a vague division of the world into 'system' on the one hand, and 'apparatus' or 'observer' on the other." 3

In particular, learning about Bohm's "hidden variables" 4 theory helped Bell recognize the invalidity of the various "no hidden variables" theorems (by John von Neumann and others) which had been taken almost universally by physicists as conclusively establishing something like Niels Bohr's Copenhagen interpretation of quantum theory. Bohm's pilot-wave theory was a clean counterexample, i.e., a proof-by-example that the theorems somehow didn't rule out what they had been taken to rule out.

This led Bell to carefully scrutinize those theorems. The result of this work was his paper "On the problem of hidden variables in quantum mechanics" 5 . This paper was written prior to the 1964 paper 6 in which Bell's theorem was first presented, but (due to an editorial accident) remained unpublished until 1966. The 1966 paper shows that the "no hidden variables" theorems of von Neumann and others all made unwarranted — and in some cases unacknowledged — assumptions. (All these theorems involved an assumption 7 which today is usually called non-contextuality .) In examining how Bohm's theory managed to violate these assumptions, Bell noticed that it did have one "curious feature": the theory was manifestly non-local. As Bell explained, "in this theory an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece." 8 This naturally raised the question of whether the non-locality was eliminable, or somehow essential:

... to the present writer's knowledge, there is no proof that any hidden variable account of quantum mechanics must have this extraordinary character. It would therefore be interesting, perhaps, to pursue some further 'impossibility proofs,' replacing the arbitrary axioms objected to above by some condition of locality, or of separability of distant systems. 9

Because of the editorial accident mentioned above, Bell had answered his own question before the paper in which it appeared was even published. The answer is contained in what we will here call "Bell's inequality theorem", which states precisely that " any hidden variable account of quantum mechanics must have this extraordinary character", i.e., must violate a locality constraint that is motivated by relativity.

But the more general result we here call "Bell's theorem" is much more than this: combined with the Einstein–Podolsky–Rosen (EPR) argument " from locality to deterministic hidden variables" 10 , the inequality theorem establishes a contradiction between locality as such (and not merely some special class of local theories) and the (now experimentally confirmed) predictions of quantum theory.

The EPR argument for pre-existing values

It is a general principle of orthodox formulations of quantum theory that measurements of physical quantities do not simply reveal pre-existing or pre-determined values, the way they do in classical theories. Instead, the particular outcome of the measurement somehow "emerges" from the dynamical interaction of the system being measured with the measuring device, so that even someone who was omniscient about the states of the system and device prior to the interaction couldn't have predicted in advance which outcome would be realized.

In a celebrated 1935 paper 11 , however, Albert Einstein, Boris Podolsky, and Nathan Rosen pointed out that, in situations involving specially-prepared pairs of particles, this orthodox principle conflicted with locality. Unfortunately, the role of locality in the discussion is often misunderstood — or missed entirely. One thus often hears that the EPR paper is essentially just an expression of (in particular) Einstein's philosophical discontent with quantum theory. This is quite wrong: what the paper actually contains is an argument showing that, if non-local influences are forbidden, and if certain quantum theoretical predictions are correct, then the measurements (whose outcomes are correlated) must be revealing pre-existing values. It is on this basis — in particular, on the assumption of locality — that EPR claimed to have established the "incompleteness" of orthodox quantum theory (which denies the existence of any such pre-existing values).

In the 1935 EPR paper, the argument was formulated in terms of position and momentum (which are observables having continuous spectra). The argument was later reformulated (by Bohm 12 ) in terms of spin. This "EPRB" version is conceptually simpler and also more closely related to the recent experiments designed to test Bell's inequality .

The EPRB argument is as follows: assume that one has prepared a pair of spin-1/2 particles in the entangled spin singlet state

with \(\left\vert\uparrow\right\rangle\ ,\) \(\left\vert\downarrow\right\rangle\) an orthonormal basis of the spin state space. A measurement of the spin of one of the particles along a given axis yields either the result "up" (i.e., "spin up") or the result "down" (i.e., "spin down"). Moreover, if one measures the spin of both particles along some given axis (say, the \(z\)-axis), then quantum theory predicts that the results obtained will be perfectly anti-correlated, i.e., they will be opposite ("up" for one particle and "down" for the other). If such measurements are carried out simultaneously on two spatially-separated particles (technically, if the measurements are performed at space-like separation) then locality requires that any disturbance triggered by the measurement on one side cannot influence the result of the measurement on the other side. But without any such interaction, the only way to ensure the perfect anti-correlation between the results on the two sides is to have each particle carry a pre-existing determinate value (appropriately anti-correlated with the value carried by the other particle) for spin along the \(z\)-axis. Any element of locally-confined indeterminism would at least sometimes spoil the predicted perfect anti-correlation between the outcomes.

Now, obviously there is nothing special here about the \(z\)-axis, so what was just established for the \(z\)-axis applies to any axis. Thus it applies to all axes at once 13 . That is, assuming (a) locality and (b) that the perfect anti-correlations predicted by quantum theory actually obtain, it follows that each particle must carry a pre-existing value for spin along all possible axes, with the values for the two particles in a given pair — which, of course, needn't be the same from one particle pair to another — perfectly anti-correlated, axis by axis. (A mathematical formulation of this argument is presented at the end of Section 5 .)

Bell's inequality theorem

Pre-existing values are thus the only local way to account for perfect anti-correlations in the outcomes of spin measurements along identical axes. But a simple argument shows that pre-existing values are incompatible with the predictions of quantum theory (for a pair of particles prepared in the singlet state) when we allow also for the possibility of spin measurements along different axes.

According to quantum theory, when spin measurements along different axes are performed on the pair of particles in the singlet state, the probability that the two results will be opposite (one "up" and one "down") is equal to \((1+\cos\,\theta)/2\ ,\) where \(\theta\in[0,\pi]\) is the angle between the chosen (oriented) axes. It follows from the simple mathematical result below, Bell's inequality theorem, that this is not compatible with the pre-existing values we have been discussing.

To see this, suppose that the spin measurements for both particles do simply reveal pre-existing values. Denote by \(Z^i_\alpha\ ,\) \(i=1,2\ ,\) the pre-determined outcome of the spin measurement for particle number \(i\) along axis \(\alpha\ .\) These values will evidently vary from one run of the experiment (i.e., one particle pair) to the next, and can thus be treated mathematically as random variables (each one assuming only two possible values, say 1 for "up" and -1 for "down").

Now consider three particular axes \(\mathbf a\ ,\) \(\mathbf b\ ,\) and \(\mathbf c\) that lie in a single plane and are such that the angle between any two of them is equal to \(2\pi/3\ .\) Then, since \(\big(1+\cos(2\pi/3)\big)/2=1/4\ ,\) agreement with quantum theory will require that \(P(Z^1_\alpha\ne Z^2_\beta)=1/4\) if \(\alpha\ne\beta\) are among \(\mathbf a\ ,\) \(\mathbf b\ ,\) \(\mathbf c\) (where \(P\) stands for probability). Agreement with quantum theory also requires opposite outcomes for identical measurement axes, i.e., \(Z^1_\alpha=-Z^2_\alpha\ ,\) for all \(\alpha\ .\) But it turns out that it is impossible to satisfy both requirements:

Bell's inequality theorem. Consider random variables \(Z^i_\alpha\ ,\) \(i=1,2\ ,\) \(\alpha=\mathbf a, \mathbf b, \mathbf c\ ,\) taking only the values \(\pm1\ .\) If these random variables are perfectly anti-correlated, i.e., if \(Z^1_\alpha=-Z^2_\alpha\ ,\) for all \(\alpha\ ,\) then: \[(1)\quad P(Z^1_{\mathbf a}\ne Z^2_{\mathbf b})+P(Z^1_{\mathbf b}\ne Z^2_{\mathbf c})+P(Z^1_{\mathbf c}\ne Z^2_{\mathbf a})\ge1.\]

Proof. Since (at any given point of the sample space) the three \(\pm1\)-valued random variables \(Z^1_\alpha\) can't all disagree, the union of the events \(\{Z^1_{\mathbf a}=Z^1_{\mathbf b}\}\ ,\) \(\{Z^1_{\mathbf b}=Z^1_{\mathbf c}\}\ ,\) \(\{Z^1_{\mathbf c}=Z^1_{\mathbf a}\}\) is equal to the entire sample space. Therefore the sum of their probabilities must be greater than or equal to 1:

\[P(Z^1_{\mathbf a}=Z^1_{\mathbf b})+P(Z^1_{\mathbf b}=Z^1_{\mathbf c})+P(Z^1_{\mathbf c}=Z^1_{\mathbf a})\ge1.\]

But since \(Z^1_\beta = -Z^2_\beta\ ,\) we have that \(P(Z^1_\alpha=Z^1_\beta)=P(Z^1_\alpha\ne Z^2_\beta)\ .\) The thesis immediately follows.

Each of the three terms on the left hand side of (1) must equal \(1/4\) in order to reproduce the quantum predictions. But, since \(1/4+1/4+1/4=3/4<1\ ,\) the full set of quantum predictions cannot be matched. This establishes the incompatibility between the quantum predictions and the existence of pre-existing values.

We note that Bell's original paper 6 considered for this purpose, instead of the disagreement probability \(P(Z^1_\alpha\ne Z^2_\beta)\ ,\) the correlation \(C(\alpha,\beta)\ ,\) defined as the expected value of the product \(Z^1_\alpha Z^2_\beta\ :\)

\[C(\alpha,\beta)=E(Z^1_\alpha Z^2_\beta)=P(Z^1_\alpha Z^2_\beta=1)\,-\,P(Z^1_\alpha Z^2_\beta=-1)=P(Z^1_\alpha=Z^2_\beta)\,-\,P(Z^1_\alpha\ne Z^2_\beta)=1\,-\,2P(Z^1_\alpha\ne Z^2_\beta).\]

Bell's original inequality (under the same assumptions as for Bell's inequality theorem above) is:

\[\vert C(\mathbf a,\mathbf b)-C(\mathbf a,\mathbf c)\vert\le 1+C(\mathbf b,\mathbf c).\]

Let us see how this inequality is related to inequality (1). Rewriting inequality (1) in terms of the correlations \(C(\alpha,\beta)\ ,\) we obtain:

\[\quad C(\mathbf a,\mathbf b)+C(\mathbf b,\mathbf c)+C(\mathbf c,\mathbf a)\le1.\]

Since (because of the perfect anti-correlations) \(C(\alpha,\beta)=C(\beta,\alpha)\ ,\) this yields that \[(2)\quad C(\mathbf a,\mathbf b)+C(\mathbf a,\mathbf c)+C(\mathbf b,\mathbf c)\le1.\]

Bell's original inequality is equivalent to the conjunction of two inequalities without absolute value: one of them is obtained from (2) by changing the signs of \(C(\mathbf a,\mathbf c)\) and \(C(\mathbf b,\mathbf c)\ .\) (This inequality follows, as (2) does, from Bell's inequality theorem above if we replace \(Z^i_{\mathbf c}\) with \(-Z^i_{\mathbf c}\ .\)) The other inequality is obtained from (2) by changing the signs of \(C(\mathbf a,\mathbf b)\) and \(C(\mathbf b,\mathbf c)\ .\) (This inequality follows from Bell's inequality theorem above by replacing \(Z^i_{\mathbf b}\) with \(-Z^i_{\mathbf b}\ .\))

Bell's theorem states that the predictions of quantum theory (for measurements of spin on particles prepared in the singlet state ) cannot be accounted for by any local theory. The proof of Bell's theorem is obtained by combining the EPR argument (from locality and certain quantum predictions to pre-existing values) and Bell's inequality theorem (from pre-existing values to an inequality incompatible with other quantum predictions).

Here is how Bell himself recapitulated the two-part argument:

Let us summarize once again the logic that leads to the impasse. The EPRB correlations are such that the result of the experiment on one side immediately foretells that on the other, whenever the analyzers happen to be parallel. If we do not accept the intervention on one side as a causal influence on the other, we seem obliged to admit that the results on both sides are determined in advance anyway, independently of the intervention on the other side, by signals from the source and by the local magnet setting. But this has implications for non-parallel settings which conflict with those of quantum mechanics. So we cannot dismiss intervention on one side as a causal influence on the other. 14

Already at the time Bell wrote this, there was a tendency for critics to miss the crucial role of the EPR argument here. The conclusion is not just that some special class of local theories (namely, those which explain the measurement outcomes in terms of pre-existing values) are incompatible with the predictions of quantum theory (which is what follows from Bell's inequality theorem alone ), but that local theories as such (whether deterministic or not, whether positing hidden variables or not, etc.) are incompatible with the predictions of quantum theory. This confusion has persisted in more recent decades, so perhaps it is worth emphasizing the point by (again) quoting from Bell's pointed footnote from the same 1980 paper quoted just above: "My own first paper on this subject ... starts with a summary of the EPR argument from locality to deterministic hidden variables. But the commentators have almost universally reported that it begins with deterministic hidden variables." 10

The CHSH–Bell inequality: Bell's theorem without perfect correlations

Perhaps motivated by this widespread and persistent misunderstanding concerning his 1964 paper 6 , Bell wrote many subsequent papers in which he explained and elaborated upon his very interesting result from a variety of angles. After 1975 15 Bell sometimes presented his result using a new strategy that does not rely on perfect (anti-)correlations and on the EPR argument. The new strategy has some advantages: perfect correlations cannot be demonstrated empirically, and one could also imagine the possibility that quantum theory might be replaced with a new theory that predicts some small deviation from the perfect correlations. So it is desirable to have a version of Bell's theorem that "depends continuously" on the correlations. The new strategy also sheds some light on the meaning of locality.

The idea is to write down a mathematically precise formulation of a consequence of locality in the context of an experiment in which measurements are performed on two systems which have previously interacted — say, systems that have been produced by a common source — but which are now spatially separated. (The EPR scenario considered above is of course an example of such an experiment.) Which of the several possible measurements are actually performed on each system will be determined by (control) parameters — \(\alpha_1\) and \(\alpha_2\) — which should be thought of as being randomly and freely chosen by the experimenters, just before the measurements. The measurements (and the choices of the control parameters) are assumed to be space-like separated. Once \(\alpha_1\) and \(\alpha_2\) are chosen, the experiment is performed, yielding (say, real-valued) outcomes \(A_1\) and \(A_2\) for the measurements on the two systems. While the values of \(A_1\) and \(A_2\) may vary from one run of the experiment to another even for the same choice of parameters, we assume that, for a fixed preparation procedure on the two systems, these outcomes exhibit statistical regularities. More precisely, we assume these are governed by probability distributions \(P_{\alpha_1,\alpha_2}(A_1,A_2)\) depending of course on the experiments performed, and in particular on \(\alpha_1\) and \(\alpha_2\ .\)

Notice that no assumption of pre-determined outcomes is being invoked here: part (or all) of the randomness of \(A_1\ ,\) \(A_2\) can arise during the process of measurement. By contrast, recall that in the above proof of Bell's inequality theorem using the random variables \(Z^i_\alpha\ ,\) the randomness was entirely located at the source, or at least occurred prior to the measurements. Moreover, in that context it was meaningful to talk about the joint probability distribution of \((Z^i_\alpha,Z^i_\beta)\) with \(\alpha\ne\beta\) (i.e., the joint probability distribution for outcomes of different measurements on the same system), while here a joint probability distribution of that type is not meaningful.

Let us now see how a mathematically precise necessary condition for locality can be formulated. First of all, one should realize that locality does not imply the independence \(P_{\alpha_1,\alpha_2}(A_1,A_2)=P_{\alpha_1,\alpha_2}(A_1)P_{\alpha_1,\alpha_2}(A_2)\) of the outcomes \(A_1\ ,\) \(A_2\ .\) Indeed, it is perfectly natural to expect that the previous interaction between the systems 1 and 2 could produce dependence relations between the outcomes. However, if locality is assumed, then it must be the case that any additional randomness that might affect system 1 after it separates from system 2 must be independent of any additional randomness that might affect system 2 after it separates from system 1. More precisely, locality requires that some set of data \(\lambda\) — made available to both systems, say, by a common source 16 — must fully account for the dependence between \(A_1\) and \(A_2\ ;\) in other words, the randomness that generates \(A_1\) out of the parameter \(\alpha_1\) and the data codified by \(\lambda\) must be independent of the randomness that generates \(A_2\) out of the parameter \(\alpha_2\) and \(\lambda\ .\) Since \(\lambda\) can vary from one run of the experiment to the other, it should be modeled as a random variable.

Let us re-state these ideas mathematically\[\lambda\] is a random variable conditioning upon which yields a decomposition

\[(3)\quad P_{\alpha_1,\alpha_2}(A_1,A_2)=\int_\Lambda P_{\alpha_1,\alpha_2}(A_1,A_2|\lambda)\,\mathrm dP(\lambda),\]

into conditional probabilities obeying a factorizability condition of the form:

\[(4)\quad P_{\alpha_1,\alpha_2}(A_1,A_2|\lambda)=P_{\alpha_1}(A_1|\lambda)P_{\alpha_2}(A_2|\lambda).\]

The probability distribution \(P\) of \(\lambda\) should not be allowed to depend on \((\alpha_1,\alpha_2)\ ;\) this is the mathematical meaning of the assumption, noted above, that the control parameters \(\alpha_1\ ,\) \(\alpha_2\) are "randomly and freely chosen by the experimenters". One might imagine here that the experimenter on each side makes a free-will choice (just before the measurement) about how to set his apparatus, that is independent of the data codified by \(\lambda\) (which existed before the choices were made). One needn't worry, however, about whether experimenters have "genuine free will" or about what that exactly means. In a real experiment, the parameters \(\alpha_1\) and \(\alpha_2\) would typically be chosen by some random or pseudo-random number generator (say, a computer) that is independent of any other physical processes that might be relevant for the outcomes, and hence independent of \(\lambda\) — unless, that is, there exists some incredible conspiracy of nature (the kind of conspiracy that would make any kind of scientific inquiry impossible). We will thus call the assumption that the probability distribution of \(\lambda\) is independent of \((\alpha_1,\alpha_2)\) the "no conspiracy" condition .

Note that the "no conspiracy" condition doesn't follow from locality: even if we assume that the choices of \(\alpha_1\) and \(\alpha_2\) are made at space-like separation from the physical processes creating the value of \(\lambda\ ,\) it is still possible in principle that the supposedly random process determining \(\alpha_1\) and \(\alpha_2\) is in fact dependent, via some local influences from the more distant past, on whatever is going on in the process that creates \(\lambda\ .\) The "no conspiracy" assumption, then, is strictly speaking just that — an additional assumption (beyond locality) on which the derivation of Bell-type inequalities rests. That said, we stress that this assumption is necessarily always made whenever one does any empirical science; in practice, one assesses the applicability of the assumption to a given experiment by examining the care with which the experimental design precludes any non-conspiratorial dependencies between the preparation of the systems and the settings of instruments 17 .

The precise mathematical setup for formulas (3) and (4) is the following: one considers a probability space \((\Lambda,P)\) and, with each \(\lambda\in\Lambda\) and each choice of the parameters \(\alpha_1\ ,\) \(\alpha_2\ ,\) one associates a probability measure \(P_{\alpha_1,\alpha_2}(\cdot|\lambda)\) on the set of possible values for the pair \((A_1,A_2)\ .\) Formula (4) says that, for each \(\lambda\in\Lambda\ ,\) the probability measure \(P_{\alpha_1,\alpha_2}(\cdot|\lambda)\) factorizes as the product of a probability measure \(P_{\alpha_1}(\cdot|\lambda)\) (the marginal of \(A_1\) given \(\lambda\)) that depends only on \(\alpha_1\) and a probability measure \(P_{\alpha_2}(\cdot|\lambda)\) (the marginal of \(A_2\) given \(\lambda\)) that depends only on \(\alpha_2\ .\) The probability distribution (3) of \((A_1,A_2)\) that is observed in the experiment (and for which quantum theory makes predictions) is obtained from \(P_{\alpha_1,\alpha_2}(\cdot|\lambda)\) by averaging (i.e., integrating) over \(\lambda\) with respect to the probability measure of the space \((\Lambda,P)\ .\) As in Section 3 , we define the correlation \(C(\alpha_1,\alpha_2)\) as the expected value of the product \(A_1A_2\) for a given choice of \(\alpha_1\ ,\) \(\alpha_2\ :\)

\[C(\alpha_1,\alpha_2)=E_{\alpha_1,\alpha_2}(A_1A_2)=\int_\Lambda E_{\alpha_1,\alpha_2}(A_1A_2|\lambda)\,\mathrm dP(\lambda),\]

where \(E_{\alpha_1,\alpha_2}(A_1A_2|\lambda)\) is the expected value of the product \(A_1A_2\) with respect to the probability measure \(P_{\alpha_1,\alpha_2}(\cdot|\lambda)\ .\)

Now it is easy to prove the CHSH inequality 18 (after John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt). This inequality is also known in the literature as the CHSH–Bell inequality or simply "Bell's inequality". In this article we will call it the "CHSH–Bell inequality" in order to distinguish it from the inequalities of Section 3 which are used in the versions of Bell's theorem that require the assumption of certain perfect (anti-)correlations.

Theorem. Suppose that the possible values for \(A_1\) and \(A_2\) are \(\pm1\ .\) Under the mathematical setup described above, assuming the factorizability condition (4), the following inequality holds:

\[|C(\mathbf a,\mathbf b)-C(\mathbf a,\mathbf c)|+|C(\mathbf a',\mathbf b)+C(\mathbf a',\mathbf c)|\le2,\]

for any choice of parameters \(\mathbf a\ ,\) \(\mathbf b\ ,\) \(\mathbf c\ ,\) \(\mathbf a'\ .\)

Proof. It follows from (4) that \(E_{\alpha_1,\alpha_2}(A_1A_2|\lambda)=E_{\alpha_1}(A_1|\lambda)E_{\alpha_2}(A_2|\lambda)\ ,\) for all \(\lambda\ ,\) \(\alpha_1\ ,\) \(\alpha_2\ .\) Thus:

\[|C(\mathbf a,\mathbf b)-C(\mathbf a,\mathbf c)|+|C(\mathbf a',\mathbf b)+C(\mathbf a',\mathbf c)|\ :\] \[\le\int_\Lambda\Big[\big|E_{\mathbf a}(A_1|\lambda)\big|\,\big(\big|E_{\mathbf b}(A_2|\lambda)-E_{\mathbf c}(A_2|\lambda)\big|\big)\,+\,\big|E_{\mathbf a'}(A_1|\lambda)\big|\,\big(\big|E_{\mathbf b}(A_2|\lambda)+E_{\mathbf c}(A_2|\lambda)\big|\big)\Big]\,\mathrm dP(\lambda)\ :\] \[\le\int_\Lambda\Big[\big|E_{\mathbf b}(A_2|\lambda)-E_{\mathbf c}(A_2|\lambda)\big|\,+\,\big|E_{\mathbf b}(A_2|\lambda)+E_{\mathbf c}(A_2|\lambda)\big|\Big]\,\mathrm dP(\lambda),\]

where the second inequality follows from the observation that \(|E_\alpha(A_1|\lambda)|\le1\ .\) The conclusion now follows directly from the following elementary lemma:

Lemma. For real numbers \(x,y\in[-1,1]\ ,\) we have that \(|x-y|+|x+y|\le2\ .\)

Proof. Squaring \(|x-y|+|x+y|\) we obtain \(2x^2+2y^2+2|x^2-y^2|\ ,\) which is either equal to \(4x^2\) or to \(4y^2\ ;\) in either case, it is less than or equal to 4.

For the experiment considered in Section 2 (spin measurements on a pair of particles in the singlet state ), quantum theory predicts \(C(\alpha,\beta)=-\alpha\cdot\beta\) (where the dot denotes the Euclidean inner product and the oriented axes \(\alpha\ ,\) \(\beta\) are identified with their corresponding unit vectors). For this experiment, the CHSH–Bell inequality is maximally violated by the quantum predictions if \(\mathbf b\) and \(\mathbf c\) are mutually orthogonal, \(\mathbf a'\) bisects \(\mathbf b\) and \(\mathbf c\ ,\) and \(\mathbf a\) bisects \(\mathbf b\) and the opposite axis \(-\mathbf c\ .\) In that case, the left hand side is equal to \(2\sqrt2\ .\) We remark also that the original Bell's inequality is obtained from the CHSH–Bell inequality by setting \(\mathbf a'=\mathbf b\) and using \(C(\mathbf b,\mathbf b)=-1\ .\)

We have thus established again the incompatibility between locality and certain predictions of quantum theory: we have proven that the CHSH–Bell inequality, which is violated by the quantum predictions, follows from the assumption of locality (and the "no conspiracy" condition).

Let us now take advantage of the mathematical formulation of (a consequence of) locality presented above — the factorizability condition (4) — in order to formulate mathematically the version of Bell's theorem presented in Section 4 . Since Bell's inequality theorem has already been formulated mathematically, it remains for us to do so for the EPR argument as well. The mathematical statement (which we will prove in a moment) corresponding to the EPR argument is the following: assuming (4) and the perfect anti-correlations \(P_{\alpha,\alpha}(A_1\ne A_2)=1\ ,\) there exist random variables \(Z^i_\alpha\) on the probability space \((\Lambda,P)\) such that:

\[(5)\quad P_{\alpha_1,\alpha_2}\big(A_i=Z^i_{\alpha_i}(\lambda)|\lambda\big)\;\stackrel{(4)}=\;P_{\alpha_i}\big(A_i=Z^i_{\alpha_i}(\lambda)|\lambda\big)=1,\]

for \(i=1,2\) and all \(\lambda\ ,\) \(\alpha_1\ ,\) and \(\alpha_2\ .\)

Notice that (using integration over \(\lambda\)) equality (5) implies that, for all \(\alpha_1\ ,\) \(\alpha_2\ ,\) the probability distribution of the pair of random variables \((Z^1_{\alpha_1},Z^2_{\alpha_2})\) is equal to the (unconditional) probability distribution (3) of the pair of outcomes \((A_1,A_2)\) (the probability distribution observed in the experiment, for which quantum theory makes predictions). In particular, we have \(P_{\alpha_1,\alpha_2}(A_1\ne A_2)=P(Z^1_{\alpha_1}\ne Z^2_{\alpha_2})\ .\) The random variables \(Z^i_\alpha\) are precisely the ingredients necessary for the proof of Bell's inequality theorem and hence we obtain, as just announced, a mathematical formulation of the version of Bell's theorem presented in Section 4.

Here is the proof of the mathematical statement corresponding to the EPR argument: assume (4) and the perfect anti-correlations. It follows from \(P_{\alpha,\alpha}(A_1\ne A_2)=1\) that \(P_{\alpha,\alpha}(A_1\ne A_2|\lambda)=1\) holds for all 19 \(\lambda\in\Lambda\ .\) When \(\alpha_1=\alpha_2=\alpha\ ,\) for each \(\lambda\in\Lambda\ ,\) the outcomes \(A_1\) and \(A_2\) given \(\lambda\) (whose joint probability distribution is \(P_{\alpha,\alpha}(\cdot|\lambda)\)) are at the same time independent (by (4)) and perfectly anti-correlated. An elementary lemma from probability theory shows that this can happen only if they are not really random, i.e., if they are constant. The constant may depend upon \(\alpha\) and \(\lambda\ ,\) and thus there are functions \(f_i\) such that \(P_{\alpha,\alpha}\big(A_i=f_i(\alpha,\lambda)|\lambda\big)=1\ .\) Define the random variables \(Z^i_\alpha\) by setting \(Z^i_\alpha(\lambda)=f_i(\alpha,\lambda)\ .\) In order to conclude the proof, observe that condition (4) implies:

\[P_{\alpha_1,\alpha_2}\big(A_i=Z^i_{\alpha_i}(\lambda)|\lambda\big)=P_{\alpha_i}\big(A_i=Z^i_{\alpha_i}(\lambda)|\lambda\big)=P_{\alpha_i,\alpha_i}\big(A_i=Z^i_{\alpha_i}(\lambda)|\lambda\big)=1.\]

Bell's definition of locality

As we have stressed above, the crucial assumption from which one can derive various empirically-testable Bell-type inequalities is locality. (Bell sometimes also used the term local causality instead of locality). Bell explained the "principle of local causality" as follows:

The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light. 20

In relativistic terms, locality is the requirement that goings-on in one region of spacetime should not affect — should not influence — happenings in space-like separated regions.

Although we have not presented any kind of careful mathematical definition of locality, we were able to prove in the previous sections that certain quantum predictions are incompatible with locality. This was achieved by means of the formulation of a mathematically precise necessary condition for locality (in the context of a particular type of experiment): namely, the factorizability condition (4). It is possible, however, to formulate locality itself in a rigorous way, at least for a certain class of physical theories. Bell actually proposed two (subtly different) such formulations, one in his 1975 paper "The theory of local beables" 15 and the other — which we will explain here — in his 1990 paper "La nouvelle cuisine" 21 .

"Beable" is Bell's term for those elements of a theory which are "to be taken seriously, as corresponding to something real" 22 . As an example, Bell cites the electric and magnetic fields of classical electromagnetism:

In Maxwell's electromagnetic theory, for example, the fields \(\mathbf E\) and \(\mathbf H\) are 'physical' (beables, we will say) but the potentials \(\mathbf A\) and \(\phi\) are 'non-physical'. Because of gauge invariance the same physical situation can be described by very different potentials. 23

As Bell points out, it is therefore no violation of locality "that in Coulomb gauge the scalar potential propagates with infinite velocity. It is not really supposed to be there." 24

The beables of a theory have values that (according to the theory) are supposed to exist independently of any observation or experiment. In this regard Bell contrasts the notion of beable with the notion of "observable" which features prominently in orthodox quantum theory:

The concept of 'observable' lends itself to very precise mathematics when identified with 'self-adjoint operator'. But physically, it is a rather woolly concept. It is not easy to identify precisely which physical processes are to be given the status of 'observations' and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the be ables, which can be described in 'classical terms', because they are there. 25

This woolliness suggests that the notion of "observation" should not appear in the formulation of (candidate) fundamental physical theories. Indeed, every aspect of a physical process (including those processes we humans classify as "observations") should be completely reducible to the actions and interactions of some physically real objects — some beables. In an "observation", both the "observed system" and the relevant experimental apparatus, for example, must be made of beables, and anything like a measurement outcome which (say) emerges anew from the system-apparatus interaction must be contained in the final disposition of those beables.

Locality is the idea that physical influences cannot propagate faster than light. It thus presupposes a clear identification, for a given candidate theory, of which elements are supposed to correspond to something that is physically real. Here is how Bell makes this point: "No one is obliged to consider the question 'What cannot go faster than light?'. But if you decide to do so, then the above remarks suggest the following: you must identify in your theory 'local be ables'" 26 . (We will discuss this again later .)

Local beables are those elements of a theory which should correspond to elements of physical reality living within spacetime . Those should include the representation of the ordinary objects of our experience, such as tables, chairs and experimental equipment. As Bell puts this:

These are the mathematical counterparts in the theory to real events at definite places and times in the real world (as distinct from the many purely mathematical constructions that occur in the working out of physical theories, as distinct from things which may be real but not localized, ...). 27

All the beables familiar from at least so-called "classical theories" are of this type — for example, the already mentioned fields in classical electromagnetism, or the positions of particles in classical mechanics. The possibility of non-local beables — corresponding to elements of physical reality which are not in spacetime — arises especially with respect to the several candidate versions of quantum theory, all of which involve a wave function (or quantum state) which, as a function on an abstract configuration space, will be a non-local beable if it is granted beable status at all. In the words of Bell:

... the wavefunction as a whole lives in a much bigger space, of \(3N\)-dimensions. It makes no sense to ask for the amplitude or phase or whatever of the wavefunction at a point in ordinary space. It has neither amplitude nor phase nor anything else until a multitude of points in ordinary three-space are specified. 28

Thus, one can meaningfully talk about "the local beables living within a region \(R\) of spacetime" or, more simply, "the local beables in region \(R\)" 29 . Those represent, according to the theory, what is supposed to be really happening in \(R\ .\) On the other hand, there is no such thing as a non-local beable "living inside" a given region of spacetime.

Not surprisingly, it is less straightforward to assess the locality of theories positing non-local beables. Let us then turn to Bell's formulation of locality (which applies straightforwardly to theories of exclusively local beables) and then return to the question of non-local beables and how Bell's formulation can be extended to apply, for example, to theories positing quantum wave functions as non-local beables.

The thought motivating Bell's formulation is that a complete specification of the physical state of (i.e., the beables in) a spacetime region which closes off the past light cone of some event should include everything needed to make predictions about that event. More precisely, such a specification should render further information — about goings-on at space-like separation from the event in question — irrelevant and/or redundant for making predictions about that event. Referring to the spacetime diagram reproduced at right, Bell formulated this as follows:

A theory will be said to be locally causal if the probabilities attached to values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a space-time region 3. 31

More precisely, the following equality of conditional probabilities must hold in a local theory: \[P(x_1|x_2,X_3)=P(x_1|X_3),\] where \(x_1\) (resp., \(x_2\)) denotes the value of a local beable in region 1 (resp., in region 2) and \(X_3\) denotes a full specification of the local beables in region 3.

As Bell goes on to explain, it is crucial that region 3 shields 32 region 1 from the overlapping past light cones of 1 and 2, and also that the specification of events in region 3 be complete; otherwise information about events in region 2 could well alter the probabilities assigned to events in 1 without this implying any violation of locality. For example, in a local non-deterministic theory, an event might occur subsequent to region 3 which was not predictable on the basis of even a complete specification of the local beables in region 3; such an event could then influence events in its own future light cone, giving rise to correlations — not predictable on the basis of information about region 3 — between space-like separated events. Such a mechanism could make information about events at space-like separation from 1 highly relevant for making predictions about 1, even when those predictions are conditionalized on complete information about region 3. The requirement that region 3 shields 1 from the overlapping past light cones of 1 and 2, however, precludes this possibility: the only way for information about such a region 2 to be relevant for predictions about 1 (once complete information about 3 has been taken into account) is if something somewhere is influencing events outside its future light cone, i.e., violating locality. It is likewise clear that information about goings-on in region 2 may very well usefully supplement predictions about events in 1 made on the basis of an incomplete specification of the values of the local beables in region 3, without any violation of locality being implied.

It is important to appreciate that Bell's proposed definition of locality applies primarily to candidate theories . There is then no particular mystery (at least for clearly-formulated theories) about, for example, which elements have beable status, or what a complete specification of local beables in some spacetime region might involve.

As suggested earlier in this section, Bell's definition of locality does not apply to arbitrary theories; also, it is not clear how one should rigorously define locality for arbitrary theories. Nevertheless, Bell's formulation can be extended in order to provide necessary conditions for the locality of the theories to which it does not apply as a definition (and, of course, such necessary conditions can be used to establish non-locality).

To begin with, Bell's definition of locality does not apply to theories positing non-local beables. Namely, one should certainly expect that not only the local beables in region 3, but also the non-local beables, should be relevant for making predictions about region 1. And of course one cannot talk about "the non-local beables in region 3" since non-local beables do not live inside regions of spacetime. However, for the only seriously-suggested example of a non-local beable — the wave function or quantum state — one can talk about its value on a Cauchy surface and it is natural (for the purpose of assessing the locality of the theory) to take "the complete description of the physical state of region 3" to mean the values of all local beables in region 3 and of the wave function in a given family of Cauchy surfaces that cover region 3.

Problems with Bell's definition also arise for non-Markovian theories, i.e., theories in which influences might "jump" over space-like surfaces. In that case, the region 3 displayed in the figure might not work properly as a shield and local 33 non-Markovian theories could be incorrectly diagnosed by Bell's definition of locality as being non-local. For the non-Markovian case, Bell's definition should then be modified so that the equality \(P(x_1|x_2,X_3)=P(x_1|X_3)\) is required to hold only when region 3 is "sufficiently thick", in some precise sense that would have to be specified, depending on how non-Markovian the theory is. In the worst case scenario, the equality \(P(x_1|x_2,X_3)=P(x_1|X_3)\) would be required to hold only when region 3 includes the entire interior of the past light cone of region 1 from some point down. We observe, however, that this modified form of Bell's definition might incorrectly diagnose some non-local theories as being local 34 so that it works only as a necessary condition for locality.

Let us now apply Bell's proposed definition of locality to the kind of experiment considered in the previous sections. (For the sake of simplicity, in what follows we will consider only theories for which Bell's definition applies directly, though it should be obvious how to adapt the exposition to more general theories for which — as discussed above — only a necessary condition for locality is available.) Recall that in Section 5 we took as a consequence of locality the factorizability condition (4); this condition involves a random variable \(\lambda\) that, by a "no conspiracy" assumption, is independent of \((\alpha_1,\alpha_2)\ .\)

Consider the spacetime diagram at right. Regions 1 and 2 contain the experiments performed on the two systems and the star in the intersection of the interior of their past light cones indicates the source. (The "particle worldlines" in the diagram are merely an illustration and play no role in the argument.) Thus, the parameter \(\alpha_1\) and the outcome \(A_1\) are (functions of) local beables in region 1 and, similarly, \(\alpha_2\) and \(A_2\) are (functions of) local beables in region 2. Note that the indicated region 3 shields off both regions 1 and 2 from their overlapping past light cones, so Bell's locality condition will require that facts about region 1 (in particular, \(\alpha_1\) and \(A_1\)) must be irrelevant for predictions about region 2, once a complete specification of the local beables in region 3 is given (and vice versa, exchanging the role of 1 and 2).

Denoting a complete specification of the local beables in region 3 by \(X\ ,\) we start with the identity 35 : \[P_{\alpha_1,\alpha_2}(A_1,A_2|X)=P_{\alpha_1,\alpha_2}(A_1|A_2,X)P_{\alpha_1,\alpha_2}(A_2|X)\] and then we use locality to obtain \(P_{\alpha_1,\alpha_2}(A_1|A_2,X)=P_{\alpha_1}(A_1|X)\) and \(P_{\alpha_1,\alpha_2}(A_2|X)=P_{\alpha_2}(A_2|X)\ .\) It follows that: \[P_{\alpha_1,\alpha_2}(A_1,A_2|X)=P_{\alpha_1}(A_1|X)P_{\alpha_2}(A_2|X).\] The equality above looks like the factorizability condition (4), but there is a difference: the variable \(X\) includes much more data than the \(\lambda\) that we considered in Section 5. While it is reasonable to assume (as a "no conspiracy" condition) that \(\lambda\) is independent of \((\alpha_1,\alpha_2)\ ,\) it is not reasonable to assume that \(X\) is independent of \((\alpha_1,\alpha_2)\ .\) Namely, since \(X\) is the complete specification of the local beables in region 3, it is not only possible but likely that \(X\) will fail to be independent of \((\alpha_1,\alpha_2)\ .\)

Of course, assuming Bell's definition of locality alone, we cannot prove the existence of a subset \(\lambda\) of the data codified by \(X\) that is independent of \((\alpha_1,\alpha_2)\) and for which (4) holds. Namely, the existence of this \(\lambda\) is not a consequence of locality alone, as it depends also on the assumption of a "no conspiracy" condition. Unlike locality, the "no conspiracy" condition involves anthropocentric elements, such as the distinction between the parameters \(\alpha_1\ ,\) \(\alpha_2\) (instrument settings, controllable by human experimenters) and the various other beables that are relevant for the experiment. For this reason, it does not seem possible to write down a clean mathematical definition of "non-conspiratorial" theory (as Bell did for local theory) in terms of conditional probabilities for values of beables posited by the theory 36 . (As usual, anthropocentric conditions are vague.) In particular, it is not possible to give a mathematical proof that for a "non-conspiratorial" local theory, there exists a \(\lambda\) independent of \((\alpha_1,\alpha_2)\) for which condition (4) holds. (Obviously, a mathematical proof cannot relate a mathematically formulated condition to a condition that is not formulated mathematically 37 .)

Nevertheless, we can argue (without any pretension to mathematical formalization) that for a "non-conspiratorial" local theory, a subset \(\lambda\) of the data codified by \(X\) satisfying these properties does exist. We do that by analyzing the meaning of various subsets of the local beables living in region 3. To begin with, notice that (it is likely that) the vast majority of those beables are irrelevant for the experiment and can be ignored. Let us then focus on the beables that are relevant for the experiment. Some of these beables (call them \(\mathfrak a_1\)) will determine or influence the setting \(\alpha_1\ .\) Similarly, some of these beables (call them \(\mathfrak a_2\)) will determine or influence the setting \(\alpha_2\ .\) One can think about \(\mathfrak a_i\) as the beables describing a computer getting ready to choose the parameter \(\alpha_i\ .\) (In a deterministic theory, the parameter \(\alpha_i\) should be a function of \(\mathfrak a_i\ ,\) but for a stochastic theory there could be additional randomness in the process that generates \(\alpha_i\) from \(\mathfrak a_i\ .\)) We take \(\lambda\) to denote the remaining local beables in region 3 that are relevant for the experiment.

For a "non-conspiratorial" theory, one must be able to define the sets of local beables \(\mathfrak a_1\ ,\) \(\mathfrak a_2\ ,\) and \(\lambda\) in such a way that \(\lambda\) is independent of \((\alpha_1,\alpha_2)\ .\) Let us now argue that, if the theory is local, condition (4) must hold for this \(\lambda\ .\) Since, among the local beables in region 3, only \(\lambda\) and \(\mathfrak a_1\) are relevant for the outcome \(A_1\) and since \(\mathfrak a_1\) is relevant to \(A_1\) only through \(\alpha_1\ ,\) the same thoughts motivating Bell's definition of locality lead to the conclusion that, upon conditioning on \(\lambda\) and \(\alpha_1\ ,\) the outcome \(A_1\) should be independent of \((A_2,\alpha_2)\ ,\) i.e., \(P_{\alpha_1,\alpha_2}(A_1|A_2,\lambda)=P_{\alpha_1}(A_1|\lambda)\ .\) For similar reasons, we have \(P_{\alpha_1,\alpha_2}(A_2|\lambda)=P_{\alpha_2}(A_2|\lambda)\) and hence \(P_{\alpha_1,\alpha_2}(A_1,A_2|\lambda)=P_{\alpha_1,\alpha_2}(A_1|A_2,\lambda)P_{\alpha_1,\alpha_2}(A_2|\lambda)=P_{\alpha_1}(A_1|\lambda)P_{\alpha_2}(A_2|\lambda)\ ,\) i.e., (4) holds.

Experiments

Bell's theorem brings out the existence of a contradiction between the empirical predictions of quantum theory and the assumption of locality. Since locality has been widely taken to be an implication of relativity theory, one thus has some grounds for wondering if the relevant predictions of quantum theory are correct. This question can only be addressed through experiment.

The first really convincing experimental tests of the relevant quantum predictions were produced in 1981—1982 by Aspect et al. 38 . These experiments involved measuring the polarizations of pairs of photons emitted (in a state of total angular momentum zero analogous to the singlet state mentioned previously) during the decay from an excited state of calcium. Correlations between the outcomes of the two polarization measurements were monitored as the axes along which the polarizations were being measured were changed. Results consistent with the quantum predictions were observed and a Bell-type inequality was violated with high statistical significance. A subsequent experiment 39 demonstrated that the quantum predictions continued to hold even when the apparatus settings (i.e., the axes along which the incoming photons' polarizations were measured) were not fixed until the last possible moment — after the photons had already been emitted by the source. (Rather than physically rotate a piece of measurement apparatus — a practical impossibility on the ten-nanosecond timescale involved in a photon's traversal of the several meters distance between the calcium source and a detector — Aspect et al. used an ingenious device that shunted each incoming photon — effectively randomly for the purpose at hand — to one of two polarization measurement devices of fixed orientation.)

The innovation of Aspect et al. represented an important first step toward closing the so-called locality loophole 40 . Recall that the locality assumption used in, for example, the derivation of the CHSH–Bell inequality , requires that the (conditional) probability distribution for possible outcomes of one of the measurements be independent of the choice of apparatus setting for the other measurement. But this is a consequence of the relativistic notion of locality only if each apparatus setting is made too late for it to affect (via influences propagating at the speed of light) the distant measurement. Fixing the final apparatus settings only after the photons (moving at the speed of light) have been emitted ensured this. However, the 1982 experiment of Aspect et al. involved, on each side of the apparatus, a periodic switching between the two possible settings (albeit with incommensurate frequencies on the two sides); one could thus conceivably still worry that the photon source and/or the nearby measurement were somehow "anticipating" the final distant apparatus setting — thus violating the formal locality assumption but without violating relativity's supposed prohibition on superluminal influences.

The locality loophole was closed much more convincingly in a more recent experiment in Innsbruck by Weihs et al. 41 in 1998. The basic experimental procedure was analogous to the one of Aspect et al. , but the Innsbruck group used entangled pairs of photons created in parametric down-conversion (instead of the decay of calcium atoms like in Aspect et al. ) and high-speed electro-optic modulators to switch between two polarization measurement settings on each side. Importantly, the modulators could be controlled on a nanosecond timescale, allowing the choice between the two possible apparatus settings on each side to be made (by independent, spatially-separated quantum random number generators) only well after the window for possible light-speed influence on the distant measurement had passed. Leaving aside the possibility of a cosmic conspiracy, this setup thus guarantees that the formal locality assumption can be violated only if some data from the measurement on one side is being somehow broadcast, faster than light, to the photon and/or measuring device on the opposite side and influencing the results there. In light of Bell's theorem the experiment thus quite conclusively establishes the relativistic non-locality of the actual world.

Other experiments (Tittel et al. 42 ) have shown that the quantum predictions remain accurate even when the particles are allowed to separate by several kilometers before their polarizations are measured. Also, in experiments designed to close the so-called detection loophole 43 (Rowe et al. 44 and Matsukevich et al. 45 ), Bell-type inequalities were violated even when a much higher fraction of all emitted pairs was successfully detected.

Another interesting recent experiment (Salart et al. 46 ) relates experimental violations of a Bell-type inequality to the motion of the earth in order to put lower limits on the speed (relative to some hypothetical preferred frame) of any involved superluminal influences.

Bell's theorem and non-contextual hidden variables

The most naive reading of standard presentations of quantum theory might lead one to the following view: the quantum observables, normally mathematically represented by self-adjoint operators on a Hilbert space, are the dynamical variables of the theory and represent elements of physical reality ( beables ). According to this view, when one talks about "measuring the observable \(A\)" one simply means that \(A\) has a value which is unknown to the experimenter and that the "measurement" makes the experimenter aware of this value (just as, say, measuring the cholesterol in my blood informs me of the pre-existing amount of cholesterol in my blood). A theory that assigns well-defined values to all quantum observables at all times (and for which "measurement" of an observable simply reveals that pre-existing assigned value) is usually known as a non-contextual hidden variables theory . There are several theorems implying that non-contextual hidden variables theories are incompatible with certain quantum predictions. (The various forms of) Bell's inequality can also be used to establish this incompatibility.

Let us explain the appropriate mathematical formulation of non-contextual hidden variables theories. Given a complex Hilbert space \(\mathcal H\ ,\) whose rays correspond to (pure) states of a quantum system, then a non-contextual hidden variables theory associates with each quantum state \(\psi\) and each self-adjoint operator \(A\) on \(\mathcal H\) a random variable \(Z_A\) on some probability space \((\Lambda,P)\) (that might depend on \(\psi\)). The value of \(Z_A\) at a point \(\lambda\) of \(\Lambda\) represents the value of the observable \(A\) for a system that, according to the theory, is described both by the quantum state \(\psi\) and by the extra variable \(\lambda\ .\) (Successive preparations of the quantum state \(\psi\) might generate different values of \(\lambda\ .\) The probability measure \(P\) on \(\Lambda\) describes their statistics.)

The compatibility condition between the non-contextual hidden variables theory and the empirical predictions of the given quantum theory is the following: if \(A_1, \ldots, A_n\) are mutually commuting self-adjoint operators on \(\mathcal H\ ,\) then the spectral measure 47 on \(\mathbb R^n\) defined from the operators \((A_1,\ldots,A_n)\) and the state \(\psi\) should coincide with the distribution of the random vector \((Z_{A_1},\ldots,Z_{A_n})\ .\) Notice that both inequality (1) and Bell's original inequality follow from the assumption of the existence of a non-contextual hidden variables theory (that covers the relevant experiments); also the CHSH–Bell inequality can be derived from this assumption. The violation of such inequalities by the quantum predictions therefore shows that non-contextual hidden variables theories are incompatible with the quantum predictions for a state space \(\mathcal H\) having at least four dimensions. (Four is, of course, the number of dimensions of the Hilbert space associated to the spin degrees of freedom of two spin-1/2 particles.) Some authors call this result "Bell's theorem" and this has given rise to a few misunderstandings .

The term "non-contextual" is motivated by the following: consider observables \(A\ ,\) \(B\) and \(C\) with \([A,B]=0\ ,\) \([A,C]=0\) but \([B,C]\ne0\ ,\) so that while \(A\) and \(B\) are jointly measurable and \(A\) and \(C\) are jointly measurable, \(B\) and \(C\) are not. (Here \([A,B]=AB-BA\) denotes the commutator of \(A\) and \(B\ .\)) Then one can perform an experiment which counts as a "measurement" of both \(A\) and \(B\) and one can also perform an experiment which counts as a "measurement" of both \(A\) and \(C\ ,\) but these experiments must be different. If one assumes that such experiments reveal pre-determined values (i.e., if one assumes that nothing truly random in the outcome is being generated by the interaction of the apparatus with the system) then, since the two experiments under consideration are different, there is no justification for assuming that the pre-determined outcomes for the "measurement" of the observable \(A\) must be the same for both experiments. More precisely: within a theory that describes the quantum system using — besides the quantum state \(\psi\) — an extra variable \(\lambda\) that determines the outcomes of experiments, one could have different functions of \(\lambda\) (for a given \(\psi\)) associated to different strategies for "measuring" an observable \(A\ .\) A non-contextual hidden variables theory is therefore one that ignores the possibility that the value assigned to \(A\) might depend on the experimental context .

In simple terms, the assumption of "non-contextuality" is the assumption that the outcome of an experiment for "measuring" an observable \(A\) does not depend on the experiment — just on the given observable. But what distinct experiments that count as "measurements" of a given observable \(A\) must have in common is only the probability distribution on the set of all possible outcomes, for every possible preparation procedure for the system on which the "measurement" is going to be performed. In other words: two different experimental arrangements \(\mathcal E\ ,\) \(\mathcal E'\) designed for "measuring" the observable \(A\) should (within a theory in which the outcomes are pre-determined) be associated to (possibly) different random variables \(Z_A^{\mathcal E}\ ,\) \(Z_A^{\mathcal E'}\) on the probability space \((\Lambda,P)\) in which \(\lambda\) takes values. Of course, agreement with the quantum predictions requires that these different random variables have the same probability distribution (for every \(\psi\)).

Since everyone knows that different random variables can have the same probability distribution, it is somewhat surprising that so many are surprised by the incompatibility between "non-contextuality" and the quantum predictions. A possible explanation for this surprise might be the fact that many quantum observables usually carry nicknames (such as "momentum" and "energy") which are motivated by their association with certain quantities that are physically real according to some classical theory from which the given quantum theory was obtained by "quantization". Of course, words such as "momentum" and "energy" are quite powerful and suggest that one is talking about some physically real quantity. However, the statement that every quantum observable corresponds to a physically real quantity that is revealed by a measurement of that observable is logically incompatible with quantum theory.

Another way to prove that non-contextual hidden variables theories are not compatible with the quantum predictions is to prove the impossibility of a value map , i.e., a map \(v\) associating with each self-adjoint operator \(A\) on \(\mathcal H\) an element \(v(A)\) of the spectrum of \(A\) in such a way that \(v(A+B)=v(A)+v(B)\) and \(v(AB)=v(A)v(B)\ ,\) whenever \(A\) and \(B\) are commuting self-adjoint operators 48 . It is easy to see that the existence of a non-contextual hidden variables theory compatible with the quantum predictions implies the existence of a value map; one must simply fix (a quantum state and) an element \(\lambda\) of the probability space \((\Lambda,P)\) where the random variables \(Z_A\) are defined and set \(v(A)=Z_A(\lambda)\) 49 . Thus, the impossibility of a value map implies the incompatibility of non-contextual hidden variables theories with the quantum predictions.

The impossibility of a value map when \(\mathrm{dim}(\mathcal H)\ge3\) follows from Gleason's theorem 50 (after Andrew M. Gleason) and also from the Kochen–Specker theorem 51 (after Simon B. Kochen and Ernst P. Specker). Another proof of the impossibility of a value map when \(\mathrm{dim}(\mathcal H)\ge3\) was given by Bell himself 5 , after Gleason and before Kochen–Specker. (See also Section IV of Mermin 52 and references therein for other proofs.) When \(\mathrm{dim}(\mathcal H)=2\ ,\) the corresponding operator algebra is somewhat trivial and it turns out that a non-contextual hidden variables theory compatible with the quantum predictions is possible; a concrete example was constructed by Bell 5 . When \(\mathrm{dim}(\mathcal H)\ge4\ ,\) a much simpler proof of the impossibility of a value map can be obtained from Mermin's theorem 53 (after David Mermin).

Bell's theorem without inequalities

There are approaches to establishing the incompatibility between locality and the quantum predictions that do not use probabilistic inequalities, but instead rely only on perfect correlations. In this section, we sketch three such approaches. The first is based on a generalization of the EPR argument given by Schrödinger; it has appeared in the general form presented here in Hemmick 54 , but particular cases of it have appeared before 55 . The second approach is based on a GHZ state 56 (after Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger) and the third approach is based on Hardy states 57 (after Lucien Hardy).

We start by presenting Hemmick's approach. It depends on the notion of maximally entangled state. Given finite-dimensional Hilbert spaces \(\mathcal H_1\ ,\) \(\mathcal H_2\) having the same dimension \(n\) and orthonormal bases \((e_1,\ldots,e_n)\ ,\) \((e'_1,\ldots,e'_n)\) of \(\mathcal H_1\) and \(\mathcal H_2\ ,\) respectively, one defines the maximally entangled state \(\psi\) associated to these bases by 58 :

If a composite system is in a maximally entangled state then to each observable \(A\) on \(\mathcal H_1\) there can be associated another observable \(\overline A\) on \(\mathcal H_2\) in such a way that a measurement of \(A\) on the system corresponding to \(\mathcal H_1\) and a measurement of \(\overline A\) on the system corresponding to \(\mathcal H_2\) must always give the same outcome 59 . We have thus a situation analogous to the one considered in the EPR argument , namely, perfect correlations between outcomes of measurements of \(A\) on the first system and outcomes of measurements of \(\overline A\) on the second system. Assuming locality and that the measurements are performed at space-like separation, we conclude that a measurement of \(A\) on the first system must actually be revealing a pre-existing value \(v(A)\ ,\) which must depend only on \(A\) and not on the experimental arrangement used to measure \(A\ .\) This map \(v\) is then a value map and any proof of the impossibility of a value map for the Hilbert space \(\mathcal H_1\) leads then to a proof of non-locality. As discussed above , such proofs can be given for \(\mathrm{dim}(\mathcal H_1)\ge3\ .\)

Let us now turn to the second approach, based on a GHZ state for three spin-1/2 particles. What we present here is a modification of a proof of the impossibility of a value map for eight-dimensional Hilbert spaces given in Mermin 60 . We consider a setup with space-like separated measurements of spin components being performed on three spin-1/2 particles. For the \(i\)-th particle, \(i=1,2,3\ ,\) the experimenter can choose between measuring spin either along the \(x\)-axis (the observable \(\sigma^i_x\)) or along the \(y\)-axis (the observable \(\sigma^i_y\)). As usual, the possible outcomes (for each particle) are either 1 or -1.

Consider the following four \(\pm1\)-valued observables:

A straightforward computation shows that these four observables are mutually commuting and that their product \(U_1U_2U_3U_4\) equals minus the identity 61 . Therefore, there exists a state \(\psi\) which is an eigenstate for all of them and, moreover, the corresponding eigenvalues \(u_1\ ,\) \(u_2\ ,\) \(u_3\ ,\) and \(u_4\) satisfy \(u_1u_2u_3u_4=-1\ .\) Assume that the state prepared by the source is this common eigenstate \(\psi\ .\)

Since \(\psi\) is an eigenstate of \(U_1\) with eigenvalue \(u_1\ ,\) if the measured observables on the three particles are chosen to be \(\sigma^1_x\ ,\) \(\sigma^2_x\ ,\) and \(\sigma^3_x\) then the product of the three outcomes obtained must be equal to \(u_1\ .\) We now use locality and a three-sided analogue of the EPR argument 62 to infer that the measurement of the observables \(\sigma^i_x\) must be revealing pre-existing values \(v(\sigma^i_x)\) satisfying \(v(\sigma^1_x)v(\sigma^2_x)v(\sigma^3_x)=u_1\ .\) Analogous arguments based on the fact that \(\psi\) is an eigenstate of the other observables \(U_2\ ,\) \(U_3\ ,\) and \(U_4\) can be used to show that the measurement of any one of the six observables \(\sigma^i_\alpha\) must be revealing a pre-existing value \(v(\sigma^i_\alpha)\ ,\) with the suitable three factor products of these pre-existing values being equal to \(u_2\ ,\) \(u_3\ ,\) and \(u_4\ .\) It follows that the product \(u_1u_2u_3u_4\) — being the square of the product of the six values \(v(\sigma^i_\alpha)\) — is equal to 1. This contradicts the fact that \(u_1u_2u_3u_4=-1\ .\)

Finally, let us turn to the third approach, based on Hardy states. Hardy states constitute a large class of entangled states for two spin-1/2 particles: namely, every entangled state that is not maximally entangled is a Hardy state. We follow the notation of Goldstein 63 , where the reader can find the detailed description of the relevant states and observables. The experimental setup consists of two spin-1/2 particles. For the \(i\)-th particle, \(i=1,2\ ,\) the experimenter can choose between measuring either the observable \(U_i\) or the observable \(W_i\ .\) The possible outcomes (for each particle) are taken to be either 0 or 1. As usual, measurements are performed at space-like separation. In what follows, for simplicity, we use the same notation for a quantum observable and for the outcome of its measurement (which, of course, is not assumed a priori to be pre-determined). For a given Hardy state, the observables \(U_i\) and \(W_i\) can be constructed so that the following four facts hold: (i) \(U_1U_2=0\ ;\) (ii) if \(U_1=0\) then \(W_2=1\ ;\) (iii) if \(U_2=0\) then \(W_1=1\ ;\) (iv) with positive probability, \(W_1=W_2=0\ .\)

It is easy to obtain a contradiction between locality and these four facts. Namely, assume locality 64 . By (i), in a given run of the experiment, either \(U_1\) or \(U_2\) must carry a pre-existing value of 0 (but it does not follow — as it does in the EPR argument — that both of them carry pre-existing values). In a given run of the experiment in which \(U_1\) carries the pre-existing value 0, it follows from (ii) that \(W_2\) must carry the pre-existing value 1. Similarly, in a given run of the experiment in which \(U_2\) carries the pre-existing value 0, it follows from (iii) that \(W_1\) must carry the pre-existing value 1. Hence, in each run of the experiment, either \(W_1\) or \(W_2\) carries the pre-existing value 1 and this contradicts (iv).

Controversy and common misunderstandings

There are many misunderstandings and controversies surrounding Bell's theorem. To begin with, we should note that while "Bell's theorem" as we have presented it here conforms with Bell's own understanding of his theorem, many other authors have presented as "Bell's theorem" very different arguments with very different conclusions — and many of those authors are often not even aware that what they are presenting differs so radically from Bell's own views. In this section we will try to shed some light on this messy state of affairs.

Missing the role of the EPR argument entirely

Section II of Bell's original paper 6 containing the celebrated theorem starts (after a short introduction, contained in the first section) with a one-paragraph recapitulation of the EPR argument (reformulated in terms of spin), i.e., it starts with the assumption of locality and it deduces from this assumption the existence of a "more complete specification of the state" (the kind of more complete specification of the state that Einstein thought would suffice to restore locality to quantum theory). Bell then claims that this more complete specification (the pre-existing values for the outcomes of spin measurements) leads to an incompatibility with the quantum predictions. The mathematical details of the proof of this incompatibility (i.e., the derivation of Bell's inequality ) appears later, in Section IV.

It seems likely that many readers didn't pay sufficient attention to the first paragraph of Section II (the beginning of Bell's argument, i.e., the EPR argument) and jumped too quickly to the mathematical considerations of Section IV (the proof of the inequality). Indeed, Bell himself comments in a footnote of a later paper that "the commentators have almost universally reported that it [his original paper] begins with deterministic hidden variables" 65 . One should also take into account the fact that, by the time Bell's theorem came along, the EPR argument was about 30 years old and it had been forgotten by many (or considered to have been somehow refuted by Bohr 66 ). Whatever the historical explanation for the misunderstanding might be, it turns out that the general understanding within the physics community regarding Bell's theorem was that it established the impossibility of "hidden variables" (or, for those a little better informed, of "local hidden variables") and the role of the EPR argument (i.e., the fact that the non-locality problem arises anyway if we regard quantum theory as complete) was missed entirely. Moreover, many authors took Bell's theorem to be a proof that, with regard to the EPR argument, Einstein was wrong and Bohr was right. While it is indeed true that Bell's theorem shows that Einstein was wrong, in that the assumption of the EPR argument (locality) turned out to be incorrect, it is not at all true that Bell's theorem shows that the EPR argument itself is not valid. In fact the EPR argument is correct and plays a crucial role in establishing that its main assumption is wrong. (That is, of course, a standard situation whenever a reductio ad absurdum is performed.)

Of course, not all commentators on Bell's theorem that disagree (knowingly or not) with Bell's conclusion have missed the EPR argument entirely. There are controversies and misunderstandings surrounding the EPR argument itself (or some poorly formulated version of it) and we shall discuss those in Subsection 10.3 . But it should be recalled that not all presentations of Bell's theorem even require the EPR argument: for instance, the CHSH–Bell inequality can be proven directly from locality, as we have shown in Section 5 . Of course, this alternative presentation of Bell's theorem generates controversies and misunderstandings of its own. One could, for instance, disagree with the claim that the mathematical formulation (4) of (a consequence of) the locality condition is adequate. (We will discuss some of those controversies and misunderstandings regarding the locality condition in Subsection 10.5 .) In fact, though, since (as we have shown ) the EPR argument becomes a simple mathematical theorem after (4) is accepted as a consequence of locality, it would be incoherent to accept that (4) is a consequence of locality but reject the EPR argument.

Bell's theorem proves the impossibility of "local realism"

One currently popular account of Bell's theorem has it showing that "local realism" is incompatible with the quantum predictions, so that one has to choose between abandoning locality or abandoning realism. Those who talk about "local realism" rarely explain what they mean by "realism". (Is "realism" related to "hidden variables" of some sort? What exactly is meant by "hidden variables"? Is "realism" related to determinism?) And when they do, it often becomes clear that the "realism" under consideration isn't among the actual assumptions of Bell's theorem, so that abandoning that kind of realism isn't a viable strategy for saving locality 67 68 . In what follows we discuss a type of "realism" that is actually relevant for Bell's theorem, but, as we will see, abandoning that kind of realism won't turn out to be a viable strategy for saving locality either.

Before we go any further, it should be pointed out that the advent of quantum theory has made many physicists quite suspicious of any analysis of what might be happening in nature when "no one is watching". The double-slit experiment, the so-called "delayed-choice" experiments, Bohr's principle of complementarity (and the EPR–Bell argument itself) are sometimes seen as evidence that certain aspects of the microscopic world transcend human understanding or, alternatively, that any discussion concerning elements of physical reality is meaningless or beyond the scope of science. (The use of the words "quantum mechanical system", Bell once noted, can have "an unfortunate effect on the discussion" 69 .) One should then allegedly settle for doing computations with operators and predicting the statistics of experimental outcomes. But, as discussed in Section 6 , the very concept of locality involved in Bell's theorem cannot even be formulated without reference to elements of physical reality, i.e., to beables (and local beables )! Unfortunately, orthodox formulations of quantum theory are notoriously vague about which (if any) variables are to be taken seriously, as beables 70 . This unfortunate situation muddles discussions regarding the locality of orthodox quantum theory.

The fact that "locality" cannot be seriously discussed without reference to local beables can be illustrated, for instance, by the following simple example: if a married man dies then his wife instantly becomes a widow. Of course, no one takes that to be an instance of non-locality. On the other hand, if the death of the husband were to cause, say, an instantaneous increase in the body temperature of his wife then this would indeed be considered a violation of locality. The difference between the two cases is that, while the state of being a widow isn't associated with any element of physical reality localized around the wife, her body temperature is: it is a function of the local beables in the region of spacetime containing the wife 71 ! Bell makes a similar point in his paper "La nouvelle cuisine":

When the Queen dies in London (may it long be delayed) the Prince of Wales, lecturing on modern architecture in Australia, becomes instantaneously King. (Greenwich Mean Time rules here.) 72

Bell goes on to present an example directly related to physics, namely, the example of the infinite velocity of propagation of the scalar potential in Coulomb gauge that we mentioned above . Bell then concludes (before he begins discussing the concept of local beable):

Conventions can propagate as fast as may be convenient. But then we must distinguish in our theory between what is convention and what is not. 73

While the EPR argument and Bell's theorem make no assumptions about what the elements of physical reality might be like, they cannot avoid talking about them. If one's criteria for accepting a sentence as being meaningful lead to the conclusion that any sentences that talk about "elements of physical reality" are meaningless 74 then, according to such criteria, the relevant notion of locality for Bell's theorem (and thus Bell's theorem itself) becomes meaningless. Those who hold that position will avoid concluding that the quantum predictions imply non-locality, but they will also avoid the conclusion that the quantum predictions are compatible with locality! So refusing to talk about elements of reality is not a strategy by which one can defend the locality of quantum theory.

Hence, one possible notion of "realism" that is actually relevant for Bell's theorem is the willingness to accept statements about elements of physical reality as in principle meaningful. This "realism" isn't, however, an independent assumption that has to be taken together with "locality" for proving Bell's theorem; it is rather a precondition for the very meaningfulness of "locality". Thus, abandoning that sort of realism does not allow one to save locality; it merely prevents one from discussing it. (Another type of "realism" that is relevant for Bell's theorem will be discussed in Subsection 10.7 .)

Some controversy regarding the EPR argument

Analyses of the EPR argument normally are focused on the presentation that appears in the original 1935 paper 11 by Einstein, Podolsky, and Rosen. The EPR paper was developed in order to present an argument establishing the incompleteness of quantum theory, i.e., establishing that there are some elements of physical reality that are omitted by the standard quantum description (in the sense that they are not determined by the quantum state 75 ).

With this goal in mind, the paper is careful about presenting a sufficient criterion for something to be an element of physical reality. The criterion presented is this: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity" 76 . This criterion simply reflects the fact that if the outcome of some experiment isn't pre-determined by some element of physical reality (i.e., if it is not a function of something that was an element of physical reality before the experiment) then its outcome involves some randomness and hence cannot be predicted with certainty. Some commentators, however, have taken Einstein's criterion to be an assumption of some sort or have objected to Einstein's use of the notion of "element of physical reality". (As discussed above , the use of such a notion could indeed conflict with someone's philosophical position regarding what sentences are to be considered meaningful.)

A somewhat different kind of criticism against the EPR argument involves the claim that it (allegedly) depends on some suspicious reasoning involving counterfactuals 77 . Here is an unfortunate formulation of the EPR argument that raises this kind of concern (under the setup with a pair of particles in the singlet state considered earlier): if the experimenter on one side chooses to measure spin along the \(z\)-axis then this experimenter can predict with certainty the outcome of the same measurement on the other side and therefore conclude that the outcome of this measurement corresponds to an element of physical reality there. The experimenter could, instead , choose to measure spin along the \(x\)-axis and, along the same lines, then conclude that the outcome of the same measurement on the other side corresponds to an element of physical reality. But the experimenter can only measure either the spin along the \(z\)-axis or the spin along the \(x\)-axis and thus (so the alleged rebuttal of the EPR argument goes) can't conclude that both the measurement outcomes (along the \(z\)-axis and along the \(x\)-axis) correspond to elements of physical reality on the other side, but rather only that one or the other (whichever one is in fact measured) does.

Considering this alleged rebuttal of the EPR argument, two observations are in order. First, for Einstein's original goal of establishing the incompleteness of quantum theory (assuming locality, of course), a simpler "single axis" version of the EPR argument is sufficient. The thesis of this "single axis" version of the EPR argument is merely that, when both experimenters choose to measure spin along the \(z\)-axis, then the outcomes of the measurements of spin along the \(z\)-axis are pre-determined. This "single axis" version of the EPR argument is (trivially) immune to the alleged rebuttal just discussed.

The second observation is that also the more general "several axes" version of the EPR argument — establishing the existence of pre-determined outcomes for measurements of spin along several axes at once — can be formulated without any counterfactuals and is therefore also immune to the alleged rebuttal discussed above. (Of course, it is this "several axes" version of the EPR argument which is needed for Bell's theorem .)

Here is the formulation of the "several axes" version of the EPR argument that does not involve counterfactuals: in order to explain (without violation of locality) the fact that the outcomes will be perfectly anti-correlated if the experimenters both measure spin along the \(z\)-axis, one has to assume that these outcomes are pre-determined. The same goes for measurements of spin along the \(x\)-axis. Even though, in each run of the experiment, either the \(z\)-axis or the \(x\)-axis is chosen along which to perform the measurements, the elements of physical reality that exist before the measurements cannot depend on choices that will be made later by the experimenters ! This, indeed, doesn't follow from the assumption of locality itself but it does follow from the so-called "no conspiracy" assumption which states, roughly speaking, that the pair of particles prepared by the source does not "know" in advance what experiments are going to be performed on them later 78 .

Classical versus quantum probability (and logic)

Some authors regard the experiments yielding a violation of Bell-type inequalities as proving that classical probability theory is wrong and that it should be replaced by quantum probability theory. The term "quantum probability" is sometimes used simply to refer to the probabilities predicted by quantum theory for outcomes of experiments; those are, of course, distinct from the probabilities predicted by, say, classical mechanics. The term is, however, more often used to refer to the theory of quantum probability spaces 79 . A quantum probability space can be defined as a pair \((\mathcal H,\psi)\) where \(\mathcal H\) is a complex Hilbert space and \(\psi\) is a unit vector in \(\mathcal H\) 80 . One then uses the term (quantum) event to refer to a closed subspace \(\mathcal S\) of \(\mathcal H\ ;\) to each such subspace one can assign a probability which is the number \(\langle \psi,P_{\mathcal S}\,\psi\rangle\in[0,1]\ ,\) where \(P_{\mathcal S}\) denotes the orthogonal projection onto \(\mathcal S\ .\) Both the set of events of a classical probability space (i.e., the \(\sigma\)-algebra of measurable subsets of the sample space) and the set of (quantum) events of a quantum probability space carry the mathematical structure of a lattice , i.e., both are partially ordered sets (in both cases the partial order is inclusion) and any pair of elements admits a least upper bound (the "or" operation) and a greatest lower bound (the "and" operation). In both cases, the greatest lower bound is the intersection while, for the classical case, the least upper bound is the union and for the quantum case it is the closure of the sum 81 .

Some formulas involving probabilities and the lattice operations of events that are true in the classical case are not true in the quantum case. This fact should not, however, be blamed on "quantum queerness" but on the fact that when one uses the words "and" and "or" to refer to the lattice operations of a quantum probability space one is using these words with non-standard meanings ! Of course, one can always change the truth value of a sentence by changing the meaning of its words and this is not evidence that physical systems are strange and counterintuitive. The motivation for calling a closed subspace \(\mathcal S\) of a Hilbert space a (quantum) event is that a "quantum measurement" of the observable \(P_{\mathcal S}\) is a \(\{0,1\}\)-valued experiment; it yields the result 1 with probability \(\langle \psi,P_{\mathcal S}\,\psi\rangle\in[0,1]\ .\) However, given two arbitrary closed subspaces \(\mathcal S_1\ ,\) \(\mathcal S_2\) of \(\mathcal H\ ,\) the \(\{0,1\}\)-valued experiments associated with \(\mathcal S_1\) and \(\mathcal S_2\) are in general mutually incompatible. Therefore a statement of the form "both the measurement of \(P_{\mathcal S_1}\) and the measurement of \(P_{\mathcal S_2}\) yield the value 1" does not correspond to any experiment and in particular is not in any way related to the experiment that is associated with the subspace \(\mathcal S_1\cap\mathcal S_2\) (except for the case in which \(P_{\mathcal S_1}\) and \(P_{\mathcal S_2}\) commute, of course).

The alleged need to abandon classical probability theory is sometimes also argued for on the basis of an incorrect analysis of the double slit experiment. However, as long as the usual meanings of words are kept, there is no need to get rid of classical probability theory (or classical logic). One should not confuse the use of the adjective "classical" as in "classical mechanics" with the use of the adjective "classical" as in "classical probability theory" or "classical logic". While classical mechanics is a physical theory which has been shown to be not empirically viable, classical probability theory and classical logic are methods of reasoning and cannot be tested empirically: such reasoning tools are what we use in order to draw conclusions from experiments so that we can decide which physical theories are or are not compatible with the results of those experiments.

Quantum probability theory is sometimes also seen as a new type of probability theory that allows for the possibility of non-commuting random variables which cannot be identified with (classical) random variables on a common probability space 82 . Of course, there is nothing "non-classical" or particularly strange about the fact that random variables on a common probability space are not always the right way to model outcomes of experiments; in fact, there is no reason why one should expect that random variables on a common probability space could be used to model the outcomes of incompatible experiments (unless one works under the assumption that the outcomes of those experiments reveal functions of elements of reality that exist independently of whether or not the experiments are performed). We will return to this point later (in Subsection 10.6 and again in Subsection 10.8 ) when we discuss again misunderstandings related to the role of non-commutativity.

Controversies and misunderstandings regarding the locality condition

The concept of locality that is relevant for Bell's theorem is sometimes mistakenly conflated with other concepts that appear in physics that are named "locality" by some authors. For instance, when one studies (classical or quantum) field theories, one learns that the Lagrangian of the theory should not contain terms of the form \(\phi(x)\phi(y)\ ,\) for example, involving the values of the field \(\phi\) at two or more different points of spacetime; Lagrangians not containing such terms are often referred to as being local . When one studies quantum field theories, one learns that space-like separated observables should commute, a requirement normally referred to as the locality condition or the local commutativity condition . Local commutativity is used to show that superluminal signalling is not possible within quantum field theory, i.e., the correlations predicted by the theory for outcomes of measurements performed at space-like separation cannot be used for communication between the experimenters. (In the notation of Section 5 , this means that the unconditional marginal distribution of the outcome \(A_1\) does not depend on the parameter \(\alpha_2\) and, similarly, the unconditional marginal of \(A_2\) does not depend on \(\alpha_1\ .\))

The locality condition for the Lagrangian, local commutativity and the impossibility of superluminal signalling are all, of course, conditions that are related to the concept of locality that is relevant for Bell's theorem. But they are not equivalent to it. In fact, the very pair correlations between observables at space-like separation on the basis of which Bell concluded that quantum mechanics is non-local are well-defined (in a frame independent way) in quantum field theory precisely because, as a consequence of local commutativity, the observables do commute.

The fact that non-locality does not imply the possibility of superluminal signalling might appear particularly surprising; this fact will seem less surprising, however, if one keeps in mind that the concept of superluminal signalling involves anthropocentric notions such as controllability and observability that play no role in the concept of locality. In simpler words, the possibility of superluminal signalling is not just non-locality, it is a form of controllable non-locality. (Notice that, for instance, while the parameters \(\alpha_i\) are controllable by the experimenters, the outcomes \(A_i\) are not.)

Other misunderstandings are reflected by certain types of objections toward the adoption of the factorizability condition (4) as a consequence of locality. For instance, one might think that the \(\lambda\) appearing in (4) is a "hidden variable" or something suspicious of that sort. Nevertheless, the \(\lambda\) could, for instance, be nothing but the quantum state (which is taken to be fixed from one run of the experiment to the other, so that the probability space \((\Lambda,P)\) in which \(\lambda\) takes values is trivial in that case). Of course, if \(\lambda\) is nothing but the quantum state then condition (4) is not satisfied by the quantum predictions, as in that case there is nothing to explain the correlation between the outcomes. (This is precisely the point raised by the EPR paper 11 .)

Some authors (notably, Jon Jarrett 83 ) have claimed that the locality condition proposed by Bell is too strong, i.e., it is more than just "locality". In order to understand the objection, one should notice first that condition (4) is equivalent to the conjunction of the following two sub-conditions:

\[(\text{OI})\quad P_{\alpha_1,\alpha_2}(A_1,A_2|\lambda)=P_{\alpha_1,\alpha_2}(A_1|\lambda)P_{\alpha_1,\alpha_2}(A_2|\lambda);\]

\[(\text{PI})\quad P_{\alpha_1,\alpha_2}(A_1|\lambda)=P_{\alpha_1}(A_1|\lambda),\quad P_{\alpha_1,\alpha_2}(A_2|\lambda)=P_{\alpha_2}(A_2|\lambda).\]

Condition (OI) says that, given \(\lambda\) and the parameters \(\alpha_1\ ,\) \(\alpha_2\ ,\) then the outcomes \(A_1\ ,\) \(A_2\) are independent. Condition (PI) says that, given \(\lambda\ ,\) then the marginal of the outcome \(A_1\) does not depend on the parameter \(\alpha_2\) and, similarly, the marginal of the outcome \(A_2\) does not depend on the parameter \(\alpha_1\ .\)

Condition (OI) is known in the literature as outcome independence and condition (PI) as parameter independence . Since the conjunction of (OI) and (PI) implies the CHSH–Bell inequality , it follows that any theory that makes the same predictions as quantum theory (and thus predicts the violation of the CHSH–Bell inequality) must violate either 84 condition (OI) or condition (PI) (or both). Some authors claim that only condition (PI), rather than (4), is a consequence of locality. Since condition (PI) doesn't have to be violated by a theory that matches the quantum predictions, these authors conclude that there is no incompatibility between the quantum predictions and locality. The misunderstanding is likely to have originated from a misinterpretation of the meaning of \(\lambda\) 85 . Namely, recall that in Section 6 we have defined \(\lambda\) to be the complete specification of the local beables (relevant for the experiment, but not for the process that chooses \(\alpha_1\) and \(\alpha_2\)) in a region of spacetime that shields the measurements from the intersection of the interior of their past lightcones 86 . It is under this particular definition for \(\lambda\) that (4) is a consequence of locality. If one takes something else as a definition of \(\lambda\) then, indeed, a violation of condition (OI) might not imply a violation of locality.

A further note concerning the conditions (OI) and (PI): the distinction between \(A_i\) and \(\alpha_i\) (which allows one to separate (4) into (OI) and (PI)) is highly anthropocentric. Namely, the parameter \(\alpha_i\) is controllable by the human experimenter and the outcome \(A_i\) isn't. But such a distinction cannot play a role in the formulation of a fundamental concept such as locality. Nevertheless, there is a difference between violation of (OI) and violation of (PI) that is worth mentioning, since it might also have caused some authors to mistakenly regard only (PI) and not (OI) as a consequence of locality: if a theory satisfies (PI) and violates (OI), then it might be the case that the kind of non-local interaction between the two sides of the experiment can be thought of as a symmetrical interaction in which there is no objective fact about which side should be regarded as the "cause" and which side should be regarded as the "effect". In fact, because of this symmetry, one might argue that the cause/effect language should not be used in this case and that one should talk only about interactions . However, if (PI) is violated, the symmetry disappears, as it is reasonable to regard a parameter \(\alpha_i\) as one of the causes of the outcome on the other side of the experiment, but it is unreasonable to regard \(\alpha_i\) as a consequence of what happened on the other side of the experiment.

Locality versus non-contextual hidden variables

We have seen in Section 5 that the CHSH–Bell inequality can be proven from the assumption of locality. It is easy to see that the CHSH–Bell inequality can also be proven from the assumption of the existence of a non-contextual hidden variables theory (one that covers the relevant experiments, of course). Namely, the assumption of non-contextual hidden variables is mathematically formulated in terms of the existence of random variables \(Z^i_{\alpha_i}\) defined over a probability space \((\Lambda,P)\) such that the outcome \(A_i\) of the experiment with parameter choice \(\alpha_i\) is equal to the value of \(Z^i_{\alpha_i}\ .\) In other words, by conditioning on a given \(\lambda\in\Lambda\ ,\) we obtain a degenerate joint probability distribution for \((A_1,A_2)\) supported by the single outcome \(\big(Z^1_{\alpha_1}(\lambda),Z^2_{\alpha_2}(\lambda)\big)\ :\) \[P_{\alpha_1,\alpha_2}\big(A_1=Z^1_{\alpha_1}(\lambda),\ A_2=Z^2_{\alpha_2}(\lambda)|\lambda\big)=1.\] This degenerate joint probability distribution of \((A_1,A_2)\) given \(\lambda\) obviously satisfies the factorizability condition (4). The purely mathematical part of the argument showing that the CHSH–Bell inequality is a consequence of locality is nothing but a proof of the CHSH–Bell inequality from condition (4). Thus, this same mathematical reasoning (restricted to the particular deterministic case in which \(A_i\) is a function of \(\alpha_i\) and \(\lambda\)) is also a proof of the CHSH–Bell inequality from the assumption of non-contextual hidden variables.

This fact has caused certain misunderstandings. To begin with, the fact that the CHSH–Bell inequality can be proven from an assumption distinct from locality leads some authors into believing that the violation of the CHSH–Bell inequality does not imply non-locality. Of course, the correct thing to say is that we have two implications:

and the logical conclusion is that the violation of the CHSH–Bell inequality by the quantum predictions gives (by (i)) yet another proof of the incompatibility of non-contextual hidden variables with the quantum predictions and (by (ii)) a proof of the incompatibility between locality and the quantum predictions.

Unfortunately, misunderstandings go beyond that. The presentation that maximizes the misunderstanding requires a different notation from what we have been using, so let us make some adaptations. Assume that the experimenter on one side can choose between measuring either the \(\pm1\)-valued quantum observable \(X_1\) or the \(\pm1\)-valued quantum observable \(Y_1\ .\) (In our old notation , the choice between \(X_1\) and \(Y_1\) corresponds to two distinct values for the parameter \(\alpha_1\ .\)) Similarly, assume that on the other side the experimenter chooses between measuring the \(\pm1\)-valued quantum observable \(X_2\) or the \(\pm1\)-valued quantum observable \(Y_2\ .\) The quantum prediction for the left hand side of the CHSH–Bell inequality — with the absolute values removed — is given by the expected value \(\langle S\rangle\) of the quantum observable 87 : \[S=X_1X_2-X_1Y_2+Y_1X_2+Y_1Y_2=X_1(X_2-Y_2)+Y_1(X_2+Y_2),\] so that the existence of a quantum state for which the CHSH–Bell inequality 88 \(\vert\langle S\rangle\vert\le2\) is violated is equivalent to the condition that the operator norm \(\Vert S\Vert\) be greater than 2. Taking into account that the observables \(X_i\ ,\) \(Y_j\) are \(\pm1\)-valued (so that their squares are equal to the identity) and that the observables carrying the index 1 commute with the observables carrying the index 2, a straightforward computation shows that: \[S^2=4+(X_1Y_1-Y_1X_1)(X_2Y_2-Y_2X_2)=4+[X_1,Y_1][X_2,Y_2].\] Since (because \(S\) is self-adjoint) \(\Vert S^2\Vert=\Vert S\Vert^2\ ,\) it follows that the existence of a quantum state for which the CHSH–Bell inequality \(\vert\langle S\rangle\vert\le2\) is violated is equivalent to the condition that \(\Vert S^2\Vert\) be greater than 4 and that condition is equivalent to the requirement that the commutators \([X_1,Y_1]\ ,\) \([X_2,Y_2]\) both be non-vanishing 89 .

Now, let us assume that we have a non-contextual hidden variables theory and, with some (here, deliberate) abuse of notation, let us use the same symbol to denote a given quantum observable and to denote the corresponding random variable given by the non-contextual hidden variables theory. Since the product of random variables is obviously commutative, the same considerations as above show that \(S^2=4\ ,\) so that \(S\) takes only the values \(\pm2\) and therefore \(\vert\langle S\rangle\vert\le2\ .\) We have thus proven again that non-contextual hidden variables imply the CHSH–Bell inequality.

The considerations above present us with the following situation: (a) there is a proof of the CHSH–Bell inequality that does not use locality (it uses non-contextual hidden variables instead) and it uses the fact that the product of random variables is commutative; (b) there is a proof that quantum theory can violate the CHSH–Bell inequality, in which non-commutativity of observables plays a prominent role; (c) it is widely believed nowadays that the great novelty of a quantum theory over a classical one is the possibility of non-commuting observables. Combining (a), (b) and (c), we can easily get the false impression that violation of the CHSH–Bell inequality has nothing to do with non-locality, but rather with a certain "non-classical" character of nature that requires "non-commuting observables". Let us put aside the fact that it is not at all clear what such a "non-classical" character of nature requiring "non-commuting observables" really means. Because of (a), (b) and (c) (combined, possibly, with other misunderstandings discussed in this article), many physicists have missed the point that one can prove the CHSH–Bell inequality from the assumption of locality alone and, therefore, no matter what one believes about the role of non-commuting observables, it follows that the violation of the CHSH–Bell inequality implies non-locality.

Many-worlds and relational interpretations of quantum theory

Strictly speaking, there is yet another assumption, besides locality and the "no conspiracy" condition that is necessary for the proof of Bell's theorem: one has to assume that, after the experiment on one given side is performed, its \(\pm1\)-valued outcome is a well-defined element of physical reality. (Recall that in Section 6 , in order to apply Bell's definition of locality to the type of experiment considered in Section 5 , we assumed that the outcomes \(A_1\) and \(A_2\) were functions of the local beables in regions 1 and 2, respectively.) Now one might wonder how anyone could deny that assumption. After all, the outcome of the experiment is recorded by the configuration of a macroscopic object (say, a pointer position, ink on a piece of paper, etc.) that can be directly inspected by a human experimenter. However, there exists one fairly popular interpretation of quantum theory that does deny that one has (after the experiments are concluded) a well-defined physically real \(\pm1\)-valued outcome on each side: the many-worlds interpretation 90 . More precisely, according to the many-worlds interpretation, both outcomes are equally real on each side, so that it doesn't make sense to talk about "the one \(\pm1\)-valued outcome that actually occurs". Certain "relational" interpretations of quantum theory 91 also deny that a completed experiment has a well-defined physically real outcome. It is possible that this type of strategy could succeed in evading the consequences of Bell's theorem, allowing for the possibility of a universe governed by a local theory such that conscious observers living in that universe attest to the validity of the quantum predictions. However, it is not clear how to actually do the trick. There are many difficulties and the subject is rather subtle. To begin with, there are controversies around the problem of finding an appropriate formulation of a many-worlds (or relational) interpretation. Moreover, it is not clear whether such an appropriate formulation can be made local, given that the wave function — which seems to be all there is in standard formulations of many-worlds theories — is not a localized object; in the terminology of Bell, it is not a local beable . (Indeed, if a theory has no local beables, it is certainly not meaningful to ask whether it is local or not in the relevant sense.) A formulation of a version of the many-worlds interpretation which includes, in addition to the wave function, some local beables, was presented in a recent paper 92 , but it was found by the authors to be non-local. The question of whether a many-worlds (or relational) approach can be taken advantage of to create a local (and empirically viable) theory thus remains open — as does the question of how seriously one should take a theory of this type, should it be successfully constructed.

Consistent histories

Proponents of the (decoherent or) consistent histories approach 93 (CH) to quantum theory claim that this approach can avoid non-locality 94 . A detailed exposition of CH is beyond the scope of this article, so let us just briefly review a few facts about it. First of all, CH is about histories; a history can be defined as a set \(\{E_1,\ldots,E_k\}\) of (orthogonal) projection operators \(E_i\ .\) A projection operator is a quantum \(\{0,1\}\)-valued observable which can be thought of as representing a proposition, such as a proposition about the "value" of a certain quantum observable (at a certain time). For example, if \(A\) is a quantum observable (i.e., a self-adjoint operator) and \(a\) is an eigenvalue of \(A\) then the proposition "\(A=a\)" is represented by the projection operator onto the \(a\)-eigenspace of \(A\) 95 .

The history \(\{E_1,\ldots,E_k\}\) is to be thought as the conjunction of the propositions \(E_1,\ldots,E_k\ .\) (It is convenient to work in the Heisenberg picture, i.e., a quantum state is considered to be fixed and time-dependence is on the observables, so that each \(E_i\) is regarded as associated with a certain instant of time.) The theory treats certain families of histories, usually known as decoherent families , as special: those are sets of histories satisfying a certain decoherence condition which is formally similar to a condition stating that certain interference terms vanish. The decoherence condition allows one to assign probabilities to the histories belonging to a decoherent family in a way that is consistent with standard rules of probability theory. The probability 96 that CH assigns to a history is simply the probability that orthodox quantum theory would assign to the observation of this history had one performed (ideal) quantum measurements of the observables \(E_i\ .\) Certain histories belong to no decoherent families at all and to such histories CH does not assign a probability. Histories consisting of mutually commuting projection operators always belong to at least one decoherent family and therefore CH defines a probability for those. (And there are also histories — belonging to some decoherent family — consisting of operators that do not commute.)

What is the main difference between CH and orthodox quantum theory? The difference is that while orthodox quantum theory is usually presented as some sort of algorithm for computing probabilities of outcomes of experiments, CH is supposed to be a theory about an objective reality in which observers making measurements do not play a privileged role. (It is supposed to be a quantum theory without observers 97 .) More precisely, a history in CH is an event that may or may not happen (even if there are no observers of any sort around) and it is to that event that the theory assigns a probability. On the other hand, orthodox quantum theory merely talks about the probability of someone observing the given history, in case the measurements of the corresponding observables \(E_i\) are actually performed. So, for instance, while orthodox quantum theory talks about the probability of someone finding a particle in a given box (using a suitable detector), CH talks about the probability of this particle being in that box (with no detector being necessary).

Thus, in CH, a "quantum measurement" is really supposed to be a measurement, simply revealing the pre-existing value of the measured observable; it is not the interaction with the apparatus that creates the observed value. That sounds a lot like a non-contextual hidden variables theory , which, as we now know, must lead to inconsistencies with the quantum predictions. Indeed, while the probabilistic statements made by CH about the histories belonging to one given decoherent family do not lead to any inconsistencies, it is very easy to show 97 (and this is uncontroversial) that probabilistic statements made by CH about histories belonging to different decoherent families sometimes do . In fact, any of the standard proofs of impossibility of non-contextual hidden variables compatible with the quantum predictions can be used to obtain such an inconsistency.

The proponents of CH have addressed this problem as follows: they have imposed a rule 98 which says essentially that arguments involving probabilities for several histories, not all of which belong to the same decoherent family, are forbidden.

To illustrate this, let us look at the EPR setup from the point of view of CH. According to CH, the spin measurements on both sides merely reveal pre-existing values and therefore there is no difficulty in locally explaining the perfect anti-correlation when the oriented axes chosen on the two sides are the same. We thus have, as the conclusion of the EPR argument asserts, random variables \(Z^i_\alpha\) as in our discussion of Bell's inequality theorem . One can now, of course, proceed to the proof of inequality (1) . CH assents that the three terms on the left hand side of (1) are equal to 1/4 and therefore we obtain the contradiction \(3/4\ge1\ .\) As explained above, this is just one of the many ways to obtain a contradiction from CH if one is allowed to use probabilities for histories belonging to different decoherent families — as do the three probabilities that appear on the left hand side of (1). However, by forbidding the reasoning used to prove inequality (1), the aforementioned rule of CH prevents us from arriving at the contradiction.

But a physical theory is not simply a game for which one can impose arbitrary rules about what reasonings are permitted for the propositions of the theory; if a physical theory implies both \(P\) and \(Q\) then the logical consequences of both \(P\) and \(Q\) will hold in a world governed by that theory and there is nothing that the proponents of the theory can do to prevent that. One might try to find an actual objection against the reasoning leading to inequality (1), but one cannot simply state as a "rule" that the reasoning is forbidden. We suspect that the proponents of CH would object to the proof of inequality (1) (within CH) by claiming that one cannot assume that all the random variables \(Z^i_\alpha\) are defined over the same probability space because on each run of the experiment the value of only one among the \(Z^1_\alpha\) and the value of only one among the \(Z^2_\alpha\) is going to be observed. But if the experiments merely reveal pre-existing values then, on each run of the experiment, all the variables \(Z^i_\alpha\) have a well-defined value (which may or may not turn out to be observed). By considering the frequencies of these (unobserved, yet existing) values, one obtains a joint probability distribution for all the variables \(Z^i_\alpha\ .\) Thus they can be modeled as random variables on the same probability space. The objection against the possibility of modeling the \(Z^i_\alpha\) as random variables on the same probability space is effective only when one takes their values to be created by the experiments: in that case, the joint probability distribution for all the \(Z^i_\alpha\) would indeed be meaningless, as their values would then correspond to the outcomes of incompatible experiments. But reinterpreted in terms of values being created by experiment, CH would be pointless — it would just be orthodox quantum theory.

Non-locality and relativity

In the previous sections, we have explained in detail the theoretical analysis according to which one can conclude from certain experimental results that non-local interactions, of the sort often thought to be precluded by relativity theory, really exist in nature. This raises the obvious question: do these experimental results, then, show that relativity is wrong ? Here we will not attempt to give an unambiguous answer, but will instead merely try to indicate very briefly some of the issues that a full answer would need to address.

To begin with, it is crucial to make a distinction between two different senses in which a theory might be said to be "relativistic". First, a theory might be empirically relativistic . This means that what it predicts for the outcomes of experiments will exhibit the usual relativistic properties — for example, it should predict the familiar relativistic behavior of clocks and meter sticks in relative motion. More generally, it should agree with classical relativistic mechanics about the behavior of macroscopic objects and it should predict (ignoring for the moment gravitation and general relativity) that an experimenter cannot tell whether an appropriately isolated laboratory has been set into uniform motion 99 .

Despite the central role it is given in certain philosophies of science, however, observation is not everything. We thus need to recognize (at least) a second sense in which a theory might be said to be "relativistic" — namely, that it is compatible with relativity through and through , and not just at the (relatively superficial) level of empirical predictions. Such a theory will be said to be fundamentally relativistic . To make the distinction clear, it is helpful to contrast two different versions of classical electromagnetism. Let us call these the Lorentzian and the Einsteinian theories.

According to the Lorentzian theory, there is a physically meaningful notion of absolute rest (defined by the so-called "ether" rest frame) and a physically meaningful notion of absolute time. These correspond to the existence of a preferred family of coordinate systems over spacetime 100 and the dynamics of the theory is defined, with respect to these coordinate systems, by the usual equations of electromagnetism. As a consequence of this dynamics, a clock at absolute rest measures absolute elapsed time, but a clock moving with absolute speed \(v\) ticks slower and it measures not absolute elapsed time, but absolute elapsed time multiplied by the usual relativistic factor \(\sqrt{1-(v/c)^2}\ .\) However, experimenters living in a world governed by this theory cannot distinguish absolute rest from absolute motion; the theory is empirically relativistic . (Bell's paper "How to teach special relativity" 101 provides a detailed discussion.)

By contrast, the Einsteinian version of classical electromagnetism is of course relativistic, not just empirically, but fundamentally . The notion of a really-existing but unobservable "ether" rest frame is dispensed with and all uniform states of motion are regarded as equivalent 102 .

Sometimes it is thought (and taught) that certain experiments from the late 19th or early 20th century refuted the Lorentzian theory in favor of the Einsteinian one. But this is not correct. With regard to their empirical predictions, there is no difference between the Lorentzian and Einsteinian theories. Nonetheless, they are different, as Bell explains in his paper "How to teach special relativity":

Since it is experimentally impossible to say which of two uniformly moving systems is really at rest, Einstein declares the notions 'really resting' and 'really moving' as meaningless. For him only the relative motion of two or more uniformly moving objects is real. Lorentz, on the other hand, preferred the view that there is indeed a state of real rest, defined by the 'aether', even though the laws of physics conspire to prevent us identifying it experimentally. The facts of physics do not oblige us to accept one philosophy rather than the other. 103

Bell suggests that a "Lorentzian pedagogy" might usefully supplement the usual approach to teaching special relativity. About this Lorentzian approach Bell writes:

Its special merit is to drive home the lesson that the laws of physics in any one reference frame account for all physical phenomena, including the observations of moving observers. And it is often simpler to work in a single frame, rather than to hurry after each moving object in turn. 104

For our purposes, there are three important lessons here. The first is that the empirical violation of Bell-type inequalities does not require theories that fail to be empirically relativistic. But this is hardly sufficient to assuage the worry: if empirical relativity is the only kind of relativity that can be saved, it's not clear that relativity, in any substantial sense, is being saved. The second lesson is thus that, if we want to insist on preserving compatibility with relativity, it is fundamental relativity (not mere empirical relativity) that we must insist on.

But the third lesson is that perhaps abandoning fundamental relativity should be on the table as a serious option. Doing so would not necessitate empirical predictions at odds with the experimental results that are normally taken to support relativity. And it is clear that the use of a dynamically preferred but unobservable "ether" frame would make it very easy for theories to incorporate the non-local interactions that Bell's theorem (and the associated experiments) require. Indeed, Bell himself took this possibility quite seriously:

It may well be that a relativistic version of [quantum] theory, while Lorentz invariant and local at the observational level, may be necessarily non-local and with a preferred frame (or aether) at the fundamental level. 105

Many readers may be puzzled by the claim that there might be any problem in making quantum theory compatible with relativity. After all, relativistic quantum field theories have been known for a long time 106 . But these theories are normally presented merely as algorithms for predicting outcomes of experiments (i.e., they are all about what observers will see when measurements are performed) and, as such, they have nothing to say about the fundamental level. Therefore, it is simply meaningless to even discuss whether or not these theories are fundamentally relativistic. The issue of fundamental relativity is meaningful only for so-called quantum theories without observers 97 , i.e., formulations of quantum theory that describe a universe in which observers play no special role in the formulation of the theory — but as a consequence of the theory are predicted to attest to the validity of the quantum predictions.

Examples of empirically (but not fundamentally) relativistic quantum theories without observers are provided by certain empirically relativistic versions of Bohmian mechanics 107 which are formulated — like the Lorentzian version of electromagnetism — in terms of a preferred family of coordinate systems corresponding to notions of absolute rest and absolute time. The fact that these theories are indeed empirically relativistic is established as follows: first, one proves that these theories make the same predictions as some empirically relativistic version of quantum theory with respect to the preferred coordinate systems in which the theory is formulated. Then, one simply appeals to the fact stated in the Bell quote mentioned above: the predictions of a given theory for one given coordinate system account for everything that happens in spacetime, including the observations of moving observers.

So, is non-locality incompatible with fundamental relativity? The main difficulty for answering this question seems to be to decide what it means for a theory to be fundamentally relativistic. Most readers might be surprised to learn that this is a non-trivial matter, since they probably can make straightforward judgements such as "(Einsteinian) Maxwell's electromagnetism is fundamentally relativistic, Newtonian mechanics is not". Of course, for many examples of physical theories it is indeed straightforward to say whether or not the given theory is fundamentally relativistic. However, it turns out not to be easy to formulate the notion of "fundamentally relativistic theory" precisely 108 and in the context of candidate theories of quantum phenomena our straightforward intuitive judgments about what a fundamentally relativistic theory should look like begin to fail us.

Let us illustrate how intuitive judgments about fundamental relativity might not be so readily available. Foliations of spacetime are normally taken to be an "anti-relativistic" structure. For instance, in the empirically (but not fundamentally) relativistic electromagnetic theory of Lorentz, a foliation — the simultaneity hyperplanes defined by the absolute time — is part of the structure of spacetime. Various empirically relativistic versions of Bohmian mechanics that are formulated in terms of a preferred family of coordinate systems corresponding to notions of absolute rest and absolute time can be reformulated by using a foliation of spacetime instead of the preferred family of coordinate systems. Incidentally, nothing forces us to interpret this foliation as being related to absolute time. What if this foliation of space-time — instead of being put in "by hand" — emerged as the solution to a Lorentz invariant law 109 ? Should the theory then be considered fundamentally relativistic? Or what if the foliation is extracted from objects — for example, the usual quantum mechanical wave function — which are already present in the theory 110 ? Or maybe the mere presence of a quantum mechanical wave function already constitutes a violation of fundamental relativity?

It is also possible that the non-local interactions required by Bell's theorem (and the associated experiments) could be incorporated without the aid of any foliation at all. Bell himself proposed a strategy 111 and in fact this strategy has led to the construction of a relativistic 112 spontaneous collapse theory of GRW type (after GianCarlo Ghirardi, Alberto Rimini and Tulio Weber 113 ) which involves no preferred foliation. Ghirardi has suggested a different strategy, based on the use of past light cones instead of a foliation 114 .

In summary, it remains unclear what exactly "fundamental relativity" means or requires. Whether Bell's theorem and the associated experiments can be reconciled with fundamental relativity thus remains very much an open question.

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Bell's Theorem

Bell's Theorem is the collective name for a family of results, all showing the impossibility of a Local Realistic interpretation of quantum mechanics. There are variants of the Theorem with different meanings of “Local Realistic.” In John S. Bell's pioneering paper of 1964 the realism consisted in postulating in addition to the quantum state a “complete state”, which determines the results of measurements on the system, either by assigning a value to the measured quantity that is revealed by the measurement regardless of the details of the measurement procedure, or by enabling the system to elicit a definite response whenever it is measured, but a response which may depend on the macroscopic features of the experimental arrangement or even on the complete state of the system together with that arrangement. Locality is a condition on composite systems with spatially separated constituents, requiring an operator which is the product of operators associated with the individual constituents to be assigned a value which is the product of the values assigned to the factors, and requiring the value assigned to an operator associated with an individual constitutent to be independent of what is measured on any other constitutent. From his assumptions Bell proved an inequality (the prototype of “Bell's Inequality”) which is violated by the Quantum Mechanical predictions made from an entangled state of the composite system. In other variants the complete state assigns probabilities to the possible results of measurements of the operators rather than determining which result will be obtained, and nevertheless inequalities are derivable; and still other variants dispense with inequalities. The incompatibility of Local Realistic Theories with Quantum Mechanics permits adjudication by experiments, some of which are described here. Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisively because of the “detection loophole” or the “communication loophole.” The latter has been nearly decisively blocked by a recent experiment and there is a good prospect for blocking the former. The refutation of the family of Local Realistic Theories would imply that certain peculiarities of Quantum Mechanics will remain part of our physical worldview: notably, the objective indefiniteness of properties, the indeterminacy of measurement results, and the tension between quantum nonlocality and the locality of Relativity Theory.

1. Introduction

2. proof of a theorem of bell's type, 3. experimental tests of bell's inequalities, 4. the detection loophole and its remedy, 5. the communication loophole and its remedy.

• 6. Variants of Bell's Theorem

7. Philosophical Comments

• 8. Appendix

Bibliography

Other internet resources, related entries.

In 1964 John S. Bell, a native of Northern Ireland and a staff member of CERN (European Organisation for Nuclear Research) whose primary research concerned theoretical high energy physics, published a paper in the short-lived journal Physics which transformed the study of the foundation of Quantum Mechanics (Bell 1964). The paper showed (under conditions which were relaxed in later work by Bell (1971, 1985, 1987) himself and by his followers (Clauser et al. 1969, Clauser and Horne 1974, Mermin 1986, Aspect 1983)) that no physical theory which is realistic and also local in a specified sense can agree with all of the statistical implications of Quantum Mechanics. Many different versions and cases, with family resemblances, were inspired by the 1964 paper and are subsumed under the italicized statement, “Bell's Theorem” being the collective name for the entire family.

One line of investigation in the prehistory of Bell's Theorem concerned the conjecture that the Quantum Mechanical state of a system needs to be supplemented by further “elements of reality” or “hidden variables” or “complete states” in order to provide a complete description, the incompleteness of the quantum state being the explanation for the statistical character of Quantum Mechanical predictions concerning the system. There are actually two main classes of hidden-variables theories. In one, which is usually called “non-contextual”, the complete state of the system determines the value of a quantity (equivalently, an eigenvalue of the operator representing that quantity) that will be obtained by any standard measuring procedure of that quantity, regardless of what other quantities are simultaneously measured or what the complete state of the system and the measuring apparatus may be. The hidden-variables theories of Kochen and Specker (1967) are explicitly of this type. In the other, which is usually called “contextual”, the value obtained depends upon what quantities are simultaneously measured and/or on the details of the complete state of the measuring apparatus. This distinction was first explicitly pointed out by Bell (1966) but without using the terms “contextual” and “non-contextual”. There actually are two quite different versions of contextual hidden-variables theories, depending upon the character of the context: an “algebraic context” is one which specifies the quantities (or the operators representing them) which are measured jointly with the quantity (or operator) of primary interest, whereas an “environmental context” is a specification of the physical characteristics of the measuring apparatus whereby it simultaneously measures several distinct co-measurable quantities. In Bohm's hidden-variables theory (1952) the context is environmental, whereas in those of Bell (1966) and Gudder (1970) the context is algebraic. A pioneering version of a “hidden variables theory” was proposed by Louis de Broglie in 1926–7 (de Broglie 1927, 1928) and a more complete version by David Bohm in 1952 (Bohm 1952; see also the entry on Bohmian mechanics ). In these theories the entity supplementing the quantum state (which is a wave function in the position representation) is typically a classical entity, located in a classical phase space and therefore characterized by both position and momentum variables. The classical dynamics of this entity is modified by a contribution from the wave function

(1)  ψ( x , t  ) = R( x , t  )exp[iS( x , t  )/ℏ],

whose temporal evolution is governed by the Schrödinger Equation. Both de Broglie and Bohm assert that the velocity of the particle satisfies the “guidance equation”

(2)  v = grad(S/m),

whereby the wave function ψ acts upon the particles as a “guiding wave”.

De Broglie (1928) and the school of “Bohmian mechanics” (notably Dürr, Goldstein, and Zanghì (1992)) postulate the guidance equation without an attempt to derive it from a more fundamental principle. Bohm (1952), however, proposes a deeper justification of the guidance equation. He postulates a modified version of Newton's second law of motion:

(3)  m d² x /d t  ² = −grad[V( x , t  ) + U( x , t  )],

where V( x , t  ) is the standard classical potential and U( x , t  ) is a new entity, the “quantum potential”,

(4)  U( x , t  ) = −(ℏ²/2m) grad²R( x , t  )/R( x , t  ),

and he proves that if the guidance equation holds at an initial time t 0 , then it follows from Eqs (2), (3), (4) and the time dependent Schrödinger Equation that it holds for all time. Although Bohm deserves credit for attempting to justify the guidance equation, there is in fact tension between that equation and the modified Newtonian equation (3), which has been analyzed by Baublitz and Shimony (1996). Eq. (3) is a second order differential equation in time and does not determine a definite solution for all t without two initial conditions — x and v at t 0 .

Since v at t 0 is a contingency, the validity of the guidance equation at t 0 (and hence at all other times) is contingent. Bohm recognizes this gap in his theory and discusses possible solutions (Bohm 1952, p. 179), without reaching a definite proposal. If, however, this difficulty is set aside, a solution is provided to the measurement problem of standard quantum mechanics, i.e., the problem of accounting for the occurrence of a definite outcome when the system of interest is prepared in a superposition of eigenstates of the operator which is subjected to measurement. Furthermore, the guidance equation ensures agreement with the statistical predictions of standard quantum mechanics. The hidden variables model using the guidance equation inspired Bell to take seriously the hidden variables interpretation of Quantum Mechanics, and the nonlocality of this model suggested his theorem.

Another approach to the hidden variables conjecture has been to investigate the consistency of the algebraic structure of the physical quantities characterized by Quantum Mechanics with a hidden variables interpretation. Standard Quantum Mechanics assumes that the “propositions” concerning a physical system are isomorphic to the lattice L ( H ) of closed linear subspaces of a Hilbert space H (equivalently, to the lattice of projection operators on H ) with the following conditions: (1) the proposition whose truth value is necessarily ‘true’ is matched with the entire space H ; (2) the proposition whose truth value is necessarily ‘false’ is matched with the empty subspace 0; (3) if a subspace S is matched with a proposition q , then the orthogonal complement of S is matched with the negation of q ; (4) the proposition q , whose truth value is ‘true’ if the truth-value of either q 1 or q 2 is ‘true’ and is ‘false’ if the answers to both q 1 and q 2 are ‘false’, is matched with the closure of the set theoretical union of the spaces S 1 and S 2 respectively matched to q 1 and q 2 , the closure being the set of all vectors which can be expressed as the sum of a vector in S 1 and a vector in S 2 . It should be emphasized that this matching does not presuppose that a proposition is necessarily either true or false and hence is compatible with the quantum mechanical indefiniteness of a truth value, which in turn underlies the feature of quantum mechanics that a physical quantity may be indefinite in value. The type of hidden variables interpretation which has been most extensively treated in the literature (often called a “non-contextual hidden variables interpretation” for a reason which will soon be apparent) is a mapping m of the lattice L into the pair {1,0}, where m ( S )=1 intuitively means that the proposition matched with S is true and m ( S )=0 means intuitively that the proposition matched with S is false. A mathematical question of importance is whether there exist such mappings for which these intuitive interpretations are maintained and conditions (1) – (4) are satisfied. A negative answer to this question for all L ( H ) where the Hilbert space H has dimensionality greater than 2 is implied by a deep theorem of Gleason (1957) (which does more, by providing a complete catalogue of possible probability functions on L ( H )). The same negative answer is provided much more simply by John Bell (1966) (but without the complete catalogue of probability functions achieved by Gleason), who also provides a positive answer to the question in the case of dimensionality 2 ; and independently these results were also achieved by Kochen and Specker (1967). It should be added that in the case of dimensionality 2 the statistical predictions of any quantum state can be recovered by an appropriate mixture of the mappings m . (See also the entry on the Kochen-Specker theorem .)

In Bell (1966), after presenting a strong case against the hidden variables program (except for the special case of dimensionality 2) Bell performs a dramatic reversal by introducing a new type of hidden variables interpretation – one in which the truth value which m assigns to a subspace S depends upon the context C of propositions measured in tandem with the one associated with S . In the new type of hidden variables interpretation the truth value into which m maps S depends upon the context C . These interpretations are commonly referred to as “contextual hidden variables interpretations”, whereas those in which there is no dependence upon the context are called “non-contextual.” Bell proves the consistency of contextual hidden variables interpretations with the algebraic structure of the lattice L ( H ) for two examples of H with dimension greater than 2. (His proposal has been systematized by Gudder (1970), who takes a context C to be a maximal Boolean subalgebra of the lattice L ( H ) of subspaces.)

Another line leading to Bell's Theorem was the investigation of Quantum Mechanically entangled states, that is, quantum states of a composite system that cannot be expressed as direct products of quantum states of the individual components. That Quantum Mechanics admits of such entangled states was discovered by Erwin Schrödinger (1926) in one of his pioneering papers, but the significance of this discovery was not emphasized until the paper of Einstein, Podolsky, and Rosen (1935). They examined correlations between the positions and the linear momenta of two well separated spinless particles and concluded that in order to avoid an appeal to nonlocality these correlations could only be explained by “elements of physical reality” in each particle — specifically, both definite position and definite momentum — and since this description is richer than permitted by the uncertainty principle of Quantum Mechanics their conclusion is effectively an argument for a hidden variables interpretation. (It should be emphasized that their argument does not depend upon counter-factual reasoning, that is reasoning about what would be observed if a quantity were measured other than the one that was in fact measured; instead their argument can be reformulated entirely in the ordinary inductive logic, as emphasized by d'Espagnat (1976) and Shimony (2001). This reformulation is important because it diminishes the force of Bohr's (1935) rebuttal of Einstein, Podolsky, and Rosen on the grounds that one is not entitled to draw conclusions about the existence of elements of physical reality from considerations of what would be seen if a measurement other than the actual one had been performed. Bell was skeptical of Bohr's rebuttal for other reasons, essentially that he regarded it to be anthropocentric. [ 1 ] See also the entry on the Einstein-Podolsky-Rosen paradox .)

At the conclusion of Bell (1966), in which Bell gives a new lease on life to the hidden variables program by introducing the notion of contextual hidden variables, he performs another dramatic reversal by raising a question about a composite system consisting of two well separated particles 1 and 2: suppose a proposition associated with a subspace S 1 of the Hilbert space of particle 1 is subjected to measurement, and a contextual hidden variables theory assigns the truth value m ( S 1 / C ) to this proposition. What physically reasonable conditions can be imposed upon the context C ? Bell suggests that C should consist only of propositions concerning particle 1, for otherwise the outcome of the measurement upon 1 will depend upon what operations are performed upon a remote particle 2, and that would be non-locality. This condition raises the question: can the statistical predictions of Quantum Mechanics concerning the entangled state be duplicated by a contextual hidden variables theory in which the context C is localized? It is interesting to note that this question also arises from a consideration of the de Broglie-Bohm model: when Bohm derives the statistical predictions of a Quantum Mechanically entangled system whose constituents are well separated, the outcome of a measurement made on one constituent depends upon the action of the “guiding wave” upon constituents that are far off, which in general will depend on the measurement arrangement on that side. Bell was thus heuristically led to ask whether a lapse of locality is necessary for the recovery of Quantum Mechanical statistics.

Bell (1964) gives a pioneering proof of the theorem that bears his name, by first making explicit a conceptual framework within which expectation values can be calculated for the spin components of a pair of spin-half particles, then showing that, regardless of the choices that are made for certain unspecified functions that occur in the framework, the expectation values obey a certain inequality which has come to be called “Bell's Inequality.” That term is now commonly used to denote collectively a family of Inequalities derived in conceptual frameworks similar to but more general than the original one of Bell. Sometimes these Inequalities are referred to as “Inequalities of Bell's type.” Each of these conceptual frameworks incorporates some type of hidden variables theory and obeys a locality assumption. The name “Local Realistic Theory” is also appropriate and will be used throughout this article because of its generality. Bell calculates the expectation values for certain products of the form (σ 1 ·â)(σ 2 ·ê), where σ 1 is the vectorial Pauli spin operator for particle 1 and σ 2 is the vectorial Pauli spin operator for particle 2 ( both particles having spin ½), and â and ê are unit vectors in three-space, and then shows that these Quantum Mechanical expectation values violate Bell's Inequality. This violation constitutes a special case of Bell's Theorem, stated in generic form in Section 1 , for it shows that no Local Realistic Theory subsumed under the framework of Bell's 1964 paper can agree with all of the statistical predictions of quantum theory.

In the present section the pattern of Bell's 1964 paper will be followed: formulation of a framework, derivation of an Inequality, demonstration of a discrepancy between certain quantum mechanical expectation values and this Inequality. However, a more general conceptual framework than his will be assumed and a somewhat more general Inequality will be derived, thus yielding a more general theorem than the one derived by Bell in 1964, but with the same strategy and in the same spirit. Papers which took the steps from Bell's 1964 demonstration to the one given here are Clauser et al. (1969), Bell (1971), Clauser and Horne (1974), Aspect (1983) and Mermin (1986). [ 2 ] Other strategies for deriving Bell-type theorems will be mentioned in Section 6 , but with less emphasis because they have, at least so far, been less important for experimental tests.

The conceptual framework in which a Bell-type Inequality will be demonstrated first of all postulates an ensemble of pairs of systems, the individual systems in each pair being labeled as 1 and 2. Each pair of systems is characterized by a “complete state” m which contains the entirety of the properties of the pair at the moment of generation. The complete state m may differ from pair to pair, but the mode of generation of the pairs establishes a probability distribution ρ which is independent of the adventures of each of the two systems after they separate. Different experiments can be performed on each system, those on 1 designated by a , a ′, etc. and those on 2 by b , b ′, etc. One can in principle let a , a ′, etc. also include characteristics of the apparatus used for the measurement, but since the dependence of the result upon the microscopic features of the apparatus is not determinable experimentally, only the macroscopic features of the apparatus (such as orientations of the polarization analyzers) in their incompletely controllable environment need be admitted in practice in the descriptions a , a ′, etc. , and likewise for b , b ′, etc. The remarkably good agreement — which will be presented in Section 3 — between the experimental measurements of correlations in Bell-type experiments and the quantum mechanical predictions of these correlations justifies restricting attention in practice to macroscopic features of the apparatus. The result of an experiment on 1 is labeled by s , which can take on any of a discrete set of real numbers in the interval [−1, 1]. Likewise the result of an experiment on 2 is labeled by t , which can take on any of a discrete set of real numbers in [−1, 1]. (Bell's own version of his theorem assumed that s and t are both bivalent, either −1 or 1, but other ranges are assumed in other variants of the theorem.)

The following probabilities are assumed to be well defined:

(5)  p 1 m ( s  | a , b , t  ) = the probability that the outcome of the measurement performed on 1 is s when m is the complete state, the measurements performed on 1 and 2 respectively are a and b , and the result of the experiment on 2 is t ; (6)  p 2 m ( t  | a , b , s  ) = the probability that the outcome of the measurement performed on 2 is t when the complete state is m , the measurements performed on 1 and 2 are respectively a and b , and the result of the measurement a is s . (7)  p m ( s , t  | a , b  ) = the probability that the results of the joint measurements a and b , when the complete state is m , are respectively s and t .

The probability function p will be assumed to be non-negative and to sum to unity when the summation is taken over all allowed values of s and t . (Note that the hidden variables theories considered in Section 1 can be subsumed under this conceptual framework by restricting the values of the probability function p to 1 and 0, the former being identified with the truth value “true” and the latter to the truth value “false”. )

A further feature of the conceptual framework is locality , which is understood as the conjunction of the following Independence Conditions :

Remote Outcome Independence (this name is a neologism, but an appropriate one, for what is commonly called outcome independence ) (8a)  p 1 m ( s  | a , b , t  ) ≡ p 1 m ( s  | a , b  ) is independent of t , (8b)  p 2 m ( t  | a , b , s  ) ≡ p 2 m ( t  | a , b  ) is independent of s ;

(Note that Eqs. (8a) and (8b) do not preclude correlations of the results of the experiment a on 1 and the experiment b on 2; they say rather that if the complete state m is given, the outcome s of the experiment on 1 provides no additional information regarding the outcome of the experiment on 2, and conversely.)

Remote Context Independence (this also is a neologism, but an appropriate one, for what is commonly called parameter independence ):

(9a)  p 1 m ( s  | a , b  ) ≡ p 1 m ( s  | a  ) is independent of b , (9b)  p 2 m ( t  | a , b  ) ≡ p 2 m ( t  | b  ) is independent of a .

Jarrett (1984) and Bell (1990) demonstrated the equivalence of the conjunction of (8a,b) and (9a,b) to the factorization condition:

(10)  p m ( s , t  | a , b  ) = p 1 m ( s  | a  ) p 2 m ( t  | b  ),

and likewise for ( a , b ′), ( a ′, b  ) , and ( a ′, b ′) substituted for ( a , b  ).

The factorizability condition Eq. (10) is also often referred to as Bell locality . It should be emphasized that at the present stage of exposition, however, Bell locality is merely a mathematical condition within a conceptual framework, to which no physical significance has been attached — in particular no connection to the locality of Special Relativity Theory, although such a connection will be made later when experimental applications of Bell's theorem will be discussed.

Bell's Inequality is derivable from his locality condition by means of a simple lemma:

(11)  If q , q ′, r , r ′ all belong to the closed interval [−1,1], then S ≡ q r + q r ′ + q ′ r − q ′ r ′ belongs to the closed interval [−2,2]. Proof: Since S is linear in all four variables q , q ′, r , r ′ it must take on its maximum and minimum values at the corners of the domain of this quadruple of variables, that is, where each of q , q ′, r , r ′ is +1 or −1. Hence at these corners S can only be an integer between −4 and +4. But S can be rewritten as ( q + q ′)( r + r ′) – 2 q ′ r ′, and the two quantities in parentheses can only be 0, 2, or −2, while the last term can only be −2 or +2, so that S cannot equal ±3, +3, −4, or +4 at the corners. Q.E.D.

Now define the expectation value of the product s t of outcomes:

(12)  E m ( a , b  ) ≡ Σ s Σ t p m ( s , t  | a , b  )( s t  ), the summation being taken over all the allowed values of s and t .

and likewise with ( a , b ′). ( a ′, b  ). and ( a ′, b ′) substituted for ( a , b  ). Also take the quantities q , q ′, r , r ′ of the above lemma (11) to be the single expectation values:

(13a)  q = Σ s   s p 1 m ( s | a ), (13b)  q ′ = Σ s   s p 1 m ( s | a ′), (13c)  r = Σ t   t p 2 m ( t  | b ), (13d)  r ′ = Σ t   t p 2 m ( t  | b ′).

Then the lemma, together with Eq. (12), factorization condition Eq. (10), and the bounds on s and t stated prior to Eq. (5), implies:

(14)  −2 ≤ E m ( a , b  ) + E m ( a , b ′) + E m ( a ′, b  ) − E m ( a ′, b ′) ≤ 2.

Finally, return to the fact that the ensemble of interest consists of pairs of systems, each of which is governed by a mapping m , but m is chosen stochastically from a space M of mappings governed by a standard probability function ρ — that is, for every Borel subset B of M , ρ( B  ) is a non-negative real number, ρ( M  ) = 1, and ρ( U j B j ) = Σ j ρ( B j ) where the B j 's are disjoint Borel subsets of M and U j B j is the set-theoretical union of the B j 's. If we define

(15a)  p ρ ( s , t | a , b  ) ≡ ∫ M p m ( s , t | a , b  ) d ρ (15b)  E ρ ( a , b  ) ≡ ∫ M E m ( a , b  ) d ρ = Σ s Σ t ∫ M p m ( s , t  | a , b  )( s t  ) d ρ

and likewise when ( a , b ′), ( a ′, b  ), and ( a ′, b ′) are substituted for ( a , b  ), then (14), (15a,b), and the properties of ρ imply

(16)  −2 ≤ E ρ ( a , b  ) + E ρ ( a , b ′) + E ρ ( a ′, b  ) − E ρ ( a ′, b ′) ≤ 2.

Ineq. (16) is an Inequality of Bell's type, henceforth called the “Bell-Clauser-Horne-Shimony-Holt (BCHSH) Inequality.”

The third step in the derivation of a theorem of Bell's type is to exhibit a system, a quantum mechanical state, and a set of quantities for which the statistical predictions violate Inequality (16). Let the system consist of a pair of photons 1 and 2 propagating in the z -direction. The two kets | x > j and | y > j constitute a polarization basis for photon j ( j =1, 2), the former representing (in Dirac's notation) a state in which the photon 1 is linearly polarized in the x -direction and the latter a state in which it is linearly polarized in the y -direction. For the two-photon system the four product kets | x > 1 | x > 2 , | x > 1 | y > 2 , | y > 1 | x > 2 , and | y > 1 | y > 2 constitute a polarization basis. Each two-photon polarization state can be expressed as a linear combination of these four basis states with complex coefficients. Of particular interest are the entangled quantum states, which in no way can be expressed as |φ> 1 |ξ> 2, with |φ> and |ξ> single-photon states, an example being

(17)  | Φ > = (1/√2)[ | x > 1 | x > 2 + | y > 1 | y > 2 ],

which has the useful property of being invariant under rotation of the x and y axes in the plane perpendicular to z . The total quantum state of the pair of photons 1 and 2 is invariant under the exchange of the two photons, as required by the fact that photons are integral spin particles. Neither photon 1 nor photon 2 is in a definite polarization state when the pair is in the state |ψ>, but their potentialities (in the terminology of Heisenberg 1958) are correlated: if by measurement or some other process the potentiality of photon 1 to be polarized along the x -direction or along the y -direction is actualized, then the same will be true of photon 2, and conversely. Suppose now that photons 1 and 2 impinge respectively on the faces of birefringent crystal polarization analyzers I and II, with the entrance face of each analyzer perpendicular to z . Each analyzer has the property of separating light incident upon its face into two outgoing non-parallel rays, the ordinary ray and the extraordinary ray . The transmission axis of the analyzer is a direction with the property that a photon polarized along it will emerge in the ordinary ray (with certainty if the crystals are assumed to be ideal), while a photon polarized in a direction perpendicular to z and to the transmission axis will emerge in the extraordinary ray. See Figure 1:

Figure 1 ( reprinted with permission )

Photon pairs are emitted from the source, each pair quantum mechanically described by |Φ> of Eq. (17), and by a complete state m if a Local Realistic Theory is assumed. I and II are polarization analyzers, with outcome s =1 and t =1 designating emergence in the ordinary ray, while s = −1 and t = −1 designate emergence in the extraordinary ray.

The crystals are also idealized by assuming that no incident photon is absorbed, but each emerges in either the ordinary or the extraordinary ray. Quantum mechanics provides an algorithm for computing the probabilities that photons 1 and 2 will emerge from these idealized analyzers in specified rays, as functions of the orientations a and b of the analyzers, a being the angle between the transmission axis of analyzer I and an arbitrary fixed direction in the x - y plane, and b having the analogous meaning for analyzer II:

(18a)  prob Φ ( s , t  | a , b  ) = | <Φ|θ s > 1 |φ t > 2 | 2 .

Here s is a quantum number associated with the ray into which photon 1 emerges, +1 indicating emergence in the ordinary ray and −1 emergence in the extraordinary ray when a is given, while t is the analogous quantum number for photon 2 when b is given; and |θ s > 1 |φ t > 2 is the ket representing the quantum state of photons 1 and 2 with the respective quantum numbers s and t . Calculation of the probabilities of interest from Eq. (18a) can be simplified by using the invariance noted after Eq. (17) and rewriting |Φ > as

(19)  |Φ> = (1/√2)[ |θ 1 > 1 |θ 1 > 2 + |θ −1 > 1 |θ −1 > 2 ].

Eq. (19) results from Eq. (17) by substituting the transmission axis of analyzer I for x and the direction perpendicular to both z and this transmission axis for y .

Since |θ −1 > 1 is orthogonal to |θ 1 > 1 , only the first term of Eq. (19) contributes to the inner product in Eq. (18a) if s = t =1; and since the inner product of | θ 1 > 1 with itself is unity because of normalization, Eq. (18a) reduces for s = t = 1 to

(18b)  prob Φ (1,1| a , b  ) = (½) | 2 <θ 1 |φ 1 > 2 | 2 .

Finally, the expression on the right hand side of Eq. (18b) is evaluated by using the law of Malus, which is preserved in the quantum mechanical treatment of polarization states: that the probability for a photon polarized in a direction n to pass through an ideal polarization analyzer with axis of transmission n ′ equals the squared cosine of the angle between n and n ′. Hence

(20a)  prob Φ (1,1| a , b  ) = (½)cos 2 σ,

where σ is b − a . Likewise,

(20b)  prob Φ (−1,−1| a , b ) = (½) cos 2 σ, and (20c)  prob Φ (1,−1| a , b ) = prob Φ (−1,1| a , b  ) = (½)sin 2 σ.

The expectation value of the product of the results s and t of the polarization analyses of photons 1 and 2 by their respective analyzers is

(21)  E Φ ( a , b  ) = prob Φ (1,1| a , b  ) + prob Φ (−1,−1| a , b  ) − prob Φ (1,−1| a , b ) − prob Φ (−1,1| a , b  ) = cos 2 σ − sin 2 σ = cos2σ.

Now choose as the orientation angles of the transmission axes

(22)  a = π/4, a ′ = 0, b = π/8, b ′ = 3 π/8 .

(23a)  E Φ ( a , b  ) = cos2(-π/8) = 0.707, (23b)  E Φ ( a , b ′) = cos2(π/8) = 0.707, (23c)  E Φ (α′, b  ) = cos2(π/8) = 0.707, (23d)  E Φ ( a ′, b ′) = cos2(3π/8) = −0.707.

(24)  S Φ ≡ E Φ ( a , b  ) + E Φ ( a , b ′) + E Φ ( a ′, b  ) − E Φ ( a ′, b ′) = 2.828.

Eq. (24) shows that there are situations where the Quantum Mechanical calculations violate the BCHSH Inequality, thereby completing the proof of a version of Bell's Theorem. It is important to note, however, that all entangled quantum states yield predictions in violation of Ineq. (16), as Gisin (1991) and Popescu and Rohrlich (1992) have independently demonstrated. Popescu and Rohrlich (1992) also show that the maximum amount of violation is achieved with a quantum state of maximum degree of entanglement, exemplified by |Φ > of Eq. (17).

In Section 3 experimental tests of Bell's Inequality and their implications will be discussed. At this point, however, it is important to discuss the significance of Bell's Theorem from a purely theoretical standpoint. What Bell's Theorem shows is that Quantum Mechanics has a structure that is incompatible with the conceptual framework within which Bell's Inequality was demonstrated: a framework in which a composite system with two subsystems 1 and 2 is described by a complete state assigning a probability to each of the possible results of every joint experiment on 1 and 2, with the probability functions satisfying the two Independence Conditions (8a,b) and (9a,b), and furthermore allowing mixtures governed by arbitrary probability functions on the space of complete states. An ‘experiment’ on a system can be understood to include the context within which a physical property of the system is measured, but the two Independence Conditions require the context to be local – that is, if a property of 1 is measured only properties of 1 co-measurable with it can be part of its context, and similarly for a property of 2. Therefore the incompatibility of Quantum Mechanics with this conceptual framework does not preclude the contextual hidden variables models proposed by Bell in (1966), an example of which is the de Broglie-Bohm model, but it does preclude models in which the contexts are required to be local. The most striking implication of Bell's Theorem is the light that it throws upon the EPR argument. That argument examines an entangled quantum state and shows that a necessary condition for avoiding action-at-a-distance between measurement outcomes of correlated properties of the two subsystems — e.g., position in both or linear momentum in both — is the ascription of “elements of physical reality” corresponding to the correlated properties to each subsystem without reference to the other. Bell's Theorem shows that such an ascription will have statistical implications in disagreement with those of quantum mechanics. A penetrating feature of Bell's analysis, when compared with that of EPR, is his examination of different properties in the two subsystems, such as linear polarizations along different directions in 1 and 2, rather than restricting his attention to correlations of identical properties in the two subsystems.

When Bell published his pioneering paper in 1964 he did not urge an experimental resolution of the conflict between Quantum Mechanics and Local Realistic Theories, probably because the former had been confirmed often and precisely in many branches of physics.

It was doubtful, however, that any of these many confirmations occurred in situations of conflict between Quantum Mechanics and Local Realistic Theories, and therefore a reliable experimental adjudication was desirable. A proposal to this effect was made by Clauser, Horne, Shimony, and Holt (1969), henceforth CHSH, who suggested that the pairs 1 and 2 be the photons produced in an atomic cascade from an initial atomic state with total angular momentum J = 0 to an intermediate atomic state with J = 1 to a final atomic state J = 0, as in an experiment performed with calcium vapor for other purposes by Kocher and Commins (1967). This arrangement has several advantages: first, by conservation of angular momentum the photon pair emitted in the cascade has total angular momentum 0, and if the photons are collected in cones of small aperture along the z -direction the total orbital angular momentum is small, with the consequence that the total spin (or polarization) angular momentum is close to 0 and therefore the polarizations of the two photons are tightly correlated; second, the photons are in the visible frequency range and hence susceptible to quite accurate polarization analysis with standard polarization analyzers; and third, the stochastic interval between the time of emission of the first photon and the time of emission of the second is in the range of 10 nsec., which is small compared to the average time between two productions of pairs, and therefore the associated photons 1 and 2 of a pair are almost unequivocally matched. A disadvantage of this arrangement, however, is that photo-detectors in the relevant frequency range are not very efficient — less than 20% efficency for single photons and hence less that 4% efficient for detection of a pair — and therefore an auxiliary assumption is needed in order to make inferences from the statistics of the subensemble of the pairs that is counted to the entire ensemble of pairs emitted during the period of observation. (This disadvantage causes a “detection loophole” which prevents the experiment, and others like it, from being decisive, but procedures for blocking this loophole are at present being investigated actively and will be discussed in Section 4 ).

In the experiment proposed by CHSH the measurements are polarization analyses with the transmission axis of analyzer I oriented at angles a and a ′, and the transmission axis of analyzer II oriented at angles b and b ′. The results s = 1 and s = −1 respectively designate passage and non-passage of photon 1 through analyzer I, and t = 1 and t = −1 respectively designate passage and non-passage of photon 2 through analyzer II. Non-passage through the analyzer is thus substituted for passage into the extraordinary ray. This simplification of the apparatus causes an obvious problem: that it is impossible to discriminate directly between a photon that fails to pass through the analyzer and one which does pass through the analyzer but is not detected because of the inefficiency of the photo-detectors. CHSH dealt with this problem in two steps. First, they expressed the probabilities p ρ ( s , t  | a , b  ), where either s or t (or both) is −1 as follows:

(25a)  p ρ (1,−1| a , b  ) = p ρ (1,1| a ,∞) − p ρ (1,1| a , b  ),

where ∞ replacing b designates the removal of the analyzer from the path of 2. Likewise

(25b)  p ρ (−1,1| a , b  ) = p ρ (1,1|∞, b  ) − p ρ (1,1| a , b  ),

where analyzer II is oriented at angle b and ∞ replacing a designates removal of the analyzer I from the path of photon 1; and finally

(25c)  p ρ (−1,−1| a , b  ) = 1 − p ρ (1,1| a , b  ) − p ρ (1,−1| a , b  ) − p ρ (−1,1| a , b  ) = 1 − p ρ (1,1| a , ∞) − p ρ (1,1| ∞, b  ) + p ρ (1,1| a , b  ),

Their second step is to make the “fair sampling assumption”: given that a pair of photons enters the pair of rays associated with passage through the polarization analyzers, the probability of their joint detection is independent of the orientation of the analyzers (including the quasi-orientation ∞ which designates removal). With this assumption, together with the assumptions that the local realistic expressions p ρ ( s , t  | a , b  ) correctly evaluate the probabilities of the results of polarization analyses of the photon pair (1,2), we can express these probabilities in terms of detection rates:

(26a)  p ρ (1,1| a , b  ) = D ( a , b  )/ D 0

where D ( a , b  ) is the counting rate of pairs when the transmission axes of analyzers I and II are oriented respectively at angles a and b , and D 0 is the detection rate when both analyzers are removed from the paths of photons 1 and 2;

(26b)  p ρ (1,1| a ,∞) = D 1 ( a ,∞)/ D 0 ,

where D 1 ( a ,∞) is the counting rate of pairs when analyzer I is oriented at angle a while analyzer II is removed; and

(26c)  p ρ (1,1|∞, b  ) = D 2 (∞, b  )/ D 0 ,

where D 2 (∞, b  ) is the counting rate when analyzer II is oriented at angle b while analyzer I is removed. When Ineq. (16) is combined with Eq. (15b) and with Eqs (26a,b,c) relating probabilities to detection rates, the result is an inequality governing detection rates,

(27)  −1 ≤ D ( a , b  )/ D 0 + D ( a , b ′)/ D 0 + D ( a ′, b  )/ D 0 − D ( a ′, b ′)/ D 0 − [ D 1 ( a )/ D 0 + D 2 ( b  )/ D 0 ] ≤ 0.

(In (27), D 1 ( a ) is an abbreviation for D ( a ,∞), and D 2 ( b ) is an abbreviation for D (∞, b ).)

If the following symmetry conditions are satisfied by the experiment:

(28)  D ( a , b  ) = D (| b − a  |), (29)  D 1 ( a  ) = D 1 , independent of a ; (30)  D 2 ( b  ) = D 2 , independent of b ,

one can rewrite Ineq. (27) as

(31)  −1 ≤ D (| b − a  |)/ D 0 + D (| b ′− a |)/ D 0 + D (| b − a ′|)/ D 0 − D (| b ′− a ′|)/ D 0 − [ D 1 / D 0 + D 2 / D 0 ] ≤ 0.

The first experimental test of Bell's Inequality, performed by Freedman and Clauser (1972), proceeded by making two applications of Ineq. (31), one to the angles a = π/4, a ′ = 0, b = π/8, b ′ = 3π/8, yielding

(32a)  −1 ≤ [3 D (π/8)/ D 0 − D (3π/8)/ D 0 ] − D 1 / D 0 − D 2 / D 0 ≤ 0,

and another to the angles a = 3π/4, a ′ = 0, b = 3π/8, b ′ = 9π/8, yielding

(32b)  −1 ≤ [3 D (3π/8)/ D 0 − D (π/8)/ D 0 ] − D 1 / D 0 − D 2 / D 0 ≤ 0.

Combining Ineq. (32a) and Ineq. (32b) yields

(33)  −(¼) ≤ [ D (π/8 )/ D 0 − D (3π/8)/ D 0 ] ≤ ¼ .

The Quantum Mechanical prediction for this arrangement, taking into account the uncertainties about the polarization analyzers and the angle from the source subtended by the analyzers, is

(34) [ D (π/8)/ D 0 − D (3π/8)/ D 0 ] Q M = (0.401+/-0.005) − (0.100+/-0.005) = 0.301+/-0.007,

The experimental result obtained by Freedman and Clauser was

(35) [ D (π/8)/ D 0 − D (3π/8)/ D 0 ] expt = 0.300 +/- 0.008,

which is 6.5 sd from the limit allowed by Ineq. (33) but in good agreement with the quantum mechanical prediction Eq. (34). This was a difficult experiment, requiring 200 hours of running time, much longer than in most later tests of Bell's Inequality, which were able to use lasers for exciting the sources of photon pairs.

Several dozen experiments have been performed to test Bell's Inequalities. References will now be given to some of the most noteworthy of these, along with references to survey articles which provide information about others. A discussion of those actual or proposed experiments which were designed to close two serious loopholes in the early Bell experiments, the “detection loophole” and the “communication loophole”, will be reserved for Section 4 and Section 5 .

Holt and Pipkin completed in 1973 (Holt 1973) an experiment very much like that of Freedman and Clauser, but examining photon pairs produced in the 9 1 P 1 →7 3 S 1 →6 3 P 0 cascade in the zero nuclear-spin isotope of mercury-198 after using electron bombardment to pump the atoms to the first state in this cascade. The result of Holt and Pipkin was in fairly good agreement with Ineq. (33), which is a consequence of the BCHSH Inequality, and in disagreement with the quantum mechanical prediction by nearly 4 sd — contrary to the results of Freedman and Clauser. Because of the discrepancy between these two early experiments Clauser (1976) repeated the Holt-Pipkin experiment, using the same cascade and excitation method but a different spin-0 isotope of mercury, and his results agreed well with the quantum mechanical predictions but violated the consequence of Bell's Inequality. Clauser also suggested a possible explanation for the anomalous result of Holt-Pipkin: that the glass of the Pyrex bulb containing the mercury vapor was under stress and hence was optically active, thereby giving rise to erroneous determinations of the polarizations of the cascade photons.

Fry and Thompson (1976) also performed a variant of the Holt-Pipkin experiment, using a different isotope of mercury and a different cascade and exciting the atoms by radiation from a narrow-bandwith tunable dye laser. Their results also agreed well with the quantum mechanical predictions and disagreed sharply with Bell's Inequality. They gathered data in only 80 minutes, as a result of the high excitation rate achieved by the laser.

Four experiments in the 1970s — by Kasday-Ullman-Wu, Faraci-Gutkowski-Notarigo-Pennisi, Wilson-Lowe-Butt, and Bruno-d’Agostino-Maroni used photon pairs produced in positronium annihilation instead of cascade photons. Of these, all but that of Faraci et al. gave results in good agreement with the quantum mechanical predictions and in disagreement with Bell's Inequalities. A discussion of these experiments is given in the review article by Clauser and Shimony (1978), who regard them as less convincing than those using cascade photons, because they rely upon stronger auxiliary assumptions.

The first experiment using polarization analyzers with two exit channels, thus realizing the theoretical scheme in the third step of the argument for Bell's Theorem in Section 2 , was performed in the early 1980s with cascade photons from laser-excited calcium atoms by Aspect, Grangier, and Roger (1982). The outcome confirmed the predictions of quantum mechanics over those of local realistic theories more dramatically than any of its predecessors — with the experimental result deviating from the upper limit in a Bell's Inequality by 40 sd. An experiment soon afterwards by Aspect, Dalibard, and Roger (1982), which aimed at closing the communication loophole, will be discussed in Section 5 . The historical article by Aspect (1992) reviews these experiments and also surveys experiments performed by Shih and Alley, by Ou and Mandel, by Rarity and Tapster, and by others, using photon pairs with correlated linear momenta produced by down-conversion in non-linear crystals. Some even more recent Bell tests are reported in an article on experiments and the foundations of quantum physics by Zeilinger (1999).

Pairs of photons have been the most common physical systems in Bell tests because they are relatively easy to produce and analyze, but there have been experiments using other systems. Lamehi-Rachti and Mittig (1976) measured spin correlations in proton pairs prepared by low-energy scattering. Their results agreed well with the quantum mechanical prediction and violated Bell's Inequality, but strong auxiliary assumptions had to be made like those in the positronium annihilation experiments. In 2003 a Bell test was performed at CERN by A. Go (Go 2003) with B-mesons, and again the results favored the quantum mechanical predictions.

The outcomes of the Bell tests provide dramatic confirmations of the prima facie entanglement of many quantum states of systems consisting of 2 or more constituents, and hence of the existence of holism in physics at a fundamental level. Actually, the first confirmation of entanglement and holism antedated Bell's work, since Bohm and Aharonov (1957) demonstrated that the results of Wu and Shaknov (1950), Compton scattering of the photon pairs produced in positronium annihilation, already showed the entanglement of the photon pairs.

The summary in Section 3 of the pioneering experiment by Freedman and Clauser noted that their symmetry assumptions, Eqs. (28), (29), and (30), are susceptible to experimental check, and furthermore could have been dispensed with by measuring the detection rates with additional orientations of the analyzers. The fair sampling assumption, on the other hand, is essential in all the optical Bell tests performed so far for linking the results of polarization or direction analysis, which are not directly observable, with detection rates, which are observable. The absence of an experimental confirmation of the fair sampling assumption, together with the difficulty of testing Bell's Inequality without this assumption or another one equally remote from confirmation is known as the “detection loophole” in the refutation of Local Realistic Theories, and is the source of skepticism about the definitiveness of the experiments.

The seriousness of the detection loophole was increased by a model of Clauser and Horne (CH) (1974), in which the rates at which the photon pairs pass through the polarization analyzers with various orientations are consistent with an Inequality of Bell's type, but the hidden variables provide instructions to the photons and the apparatus not only regarding passage through the analyzers but also regarding detection, thereby violating the fair sampling assumption. Detection or non-detection is selective in the model in such a way that the detection rates violate the Bell-type Inequality and agree with the quantum mechanical predictions. Other models were constructed later by Fine (1982a) and corrected by Maudlin (1994) (the “Prism Model”) and by C.H.Thompson (1996) (the “Chaotic Ball model”). Although all these models are ad hoc and lack physical plausibility, they constitute existence proofs that Local Realistic Theories can be consistent with the quantum mechanical predictions provided that the detectors are properly selective. The detection loophole could in principle be blocked by a test of the BCHSH Inequality provided that the detectors for the 1 and 2 particles were sufficiently efficient and that there were a reliable way of determining how many pairs impinge upon the analyzer-detector assemblies. The first of these conditions can very likely be fulfilled if atoms from the dissociation of dimers are used as the particle pairs, as in the proposed experiment of Fry and Walther (Fry & Walther 1997, 2002, Walther & Fry 1997), to be discussed below, but the second condition does not at present seem feasible. Consequently a different strategy is needed.

Tools which are promising for blocking the detection loophole are two Inequalities derived by CH (Clauser & Horne 1974), henceforth called BCH Inequalities. Both differ from the BCHSH Inequality of Section 2 by involving ratios of probabilities. The two BCH Inequalities differ from each other in two respects: the first involves not only the probabilities of coincident counting of the 1 and 2 systems but also probabilities of single counting of 1 and 2 without reference to the other, and it makes no auxiliary assumption regarding the dependence of detection upon the placement of the analyzer; the second involves only probabilities of coincident detection and uses a weak, but still non-trivial, auxiliary assumption called “no enhancement.”

The experiment proposed by Fry and Walther intends to make use of the first BCH Inequality, dispensing with auxiliary assumptions, but the second BCH Inequality will also be reviewed here, because of its indispensability in case the optimism of Fry and Walther regarding the achievability of certain desirable experimental conditions turns out to be disappointed.

The conceptual framework of both BCH Inequalities takes an analyzer/detector assembly as a unit, instead of considering the separate operation of two parts, and it places only one analyzer/detector assembly in the 1 arm and one in the 2 arm of the experiment; in other words, for both analyzers I and II the two exit channels are detection and non-detection. Let N be the number of pairs of systems impinging on the two analyzer/detector assemblies, N 1 (a) the number detected by 1's analyzer/detector assembly, N 2 (b) the number detected by 2's analyzer/detector assembly, and N 12 ( a , b  ) the number of pairs detected by both, where a and b are the respective settings of the analyzers. Let N 1 ( m , a  ), N 2 ( m , b  ), and N 12 ( m , a , b  ) be the corresponding quantities predicted by the Local Realistic Theory with complete state m . Then the respective probabilities of detection by the analyzer/detector assemblies for 1 and 2 separately and by both together predicted by the local realistic theory with complete state m are

(36a)  p 1 ( m , a  ) = N 1 ( m , a  )/ N , (36b)  p 2 ( m , b  ) = N 2 ( m , b  )/ N (36c)  p ( m , a , b  ) = N 12 ( m , a , b  )/ N .

Note that the total number N of pairs appears in the denominators on the right hand side, but the difficulty of determining this quantity experimentally is circumvented by CH, who derive an inequality concerning the ratios of the probabilities in Eqs. (36a,b,c), so that the denominator N cancels. The analogues of the Independence Conditions of (8a,b) and (9a,b) are taken as part of the conceptual framework of a Local Realistic Theory, which implies the factorization of p ( m , a , b  ):

(37)  p ( m , a , b  ) = p 1 ( m , a  ) p 2 ( m , b  ).

CH then prove a lemma similar to the lemma in Section 2 : if q , q ′ are real numbers such that q and q ′ fall in the closed interval [0, X ], and r , r ′,are real numbers such that r and r ′ fall in the closed interval [0, Y ], then

(38)  −1 ≤ q r + q r ′ + q ′ r − q ′ r ′ − q Y − r X ≤ 0.

Then making the substitution

(39)  q = p 1 ( m , a  ), q ′ = p 1 ( m , a ′), r = p 2 ( m , b  ), r ′ = p 2 ( m , b ′),

and using Eq. (37) we have

(40)  −1 ≤ p ( m , a , b  ) + p ( m , a , b ′) + p ( m , a ′, b  ) − p ( m , a ′, b ′) − p 1 ( m , a  ) Y − p 2 ( m , b  ) X ≤ 0.

When the Local Realistic Theory describes an ensemble of pairs by a probability distribution ρ over the space of complete states M then the probabilities corresponding to those of (36a,b,c) are

(41a)  p ρ 1 (a) = ∫ M p 1 ( m , a  ) d ρ,

(41b)  p ρ 2 ( b  ) = ∫ M p 2 ( m , b  ) d ρ,

(41c)  p ρ ( a , b  ) = ∫ M p ( m , a , b  ) d ρ,

and when X and Y are taken to be 1, as they certainly can be from general properties of probability distributions, then

(42)  −1 ≤ p ρ ( a , b  ) + p ρ ( a , b ′) + p ρ ( a ′, b  ) − p ρ ( a ′, b ′) − p ρ 1 ( a  ) − p ρ 2 ( b  ) ≤ 0 ,

Since CH are seeking an Inequality involving only the ratios of probabilities they disregard the lower limit and rewrite the right hand Inequality as

(43)  [ p ρ ( a , b  ) + p ρ ( a , b ′) + p ρ ( a ′, b  ) − p ρ ( a ′, b ′)]/[ p ρ 1 ( a  ) + p ρ 2 ( b  )] ≤ 1.

This is the first BCH Inequality. In principle this Inequality could be used to adjudicate between the family of Local Realistic Theories and Quantum Mechanics, provided that the detectors are sufficiently efficient and also provided that the single detection counts are not spoiled by counting systems that do not belong to the pairs in the ensemble of interest. In experiments using photon pairs from a cascade, as most of the early Bell tests were, it can happen that the second transition occurs without the first step in the cascade, thus producing a single photon without a partner. To cope with this difficulty CH make the “no enhancement assumption”, which is considerably weaker than the “fair sampling” assumption used in Section 3 : that if an analyzer is removed from the path of either 1 or 2 — an operation designated symbolically by letting ∞ replace the parameter a or b of the respective analyzer — the resulting probability of detection is at least as great as when a finite parameter is used, i.e. ,

(44a)  p 1 ( m , a  ) ≤ p 1 ( m ,∞), (44b)  p 2 ( m , b  ) ≤ p 2 ( m ,∞)

Now let the right hand side of (44a) be X and the right hand side of (44b) be Y and insert these values into Inequality (40), and furthermore use Eq. (37) twice to obtain

(45)  −1 ≤ p ( m , a , b  ) + p ( m , a , b ′) + p ( m , a ′, b  ) − p ( m , a ′, b ′) − p 1 ( m , a ,∞) − p 2 ( m ,∞, b  ) ≤ 0.

Integrate Inequality (45) using the distribution ρ and then retain only the right hand Inequality in order to obtain an expression involving ratios only of probabilities of joint detections:

(46)  [ p ρ ( a , b  ) + p ρ ( a , b ′) + p ρ ( a ′, b  ) − p ρ ( a ′, b ′)] / [ p ρ ( a ,∞) + p ρ (∞, b  )] ≤ 1

This is the second BCH Inequality.

In the experiment initiated by Fry and Walther (1997), but not yet complete, dimers of 199 Hg are generated in a supersonic jet expansion and then photo-dissociated by two photons from appropriate laser beams. Each 199 Hg atom has nuclear spin ½, and the total spin F (electronic plus nuclear) of the dimer is 0 because of the symmetry rules for the total wave function of a homonuclear diatomic molecule consisting of two fermions. (The argument for this conclusion is fairly intricate but well presented in Walther and Fry 1997). Because the dissociation is a two-body process, momentum conservation guarantees that the directions of the two atoms after dissociation are strictly correlated, so that when the two analyzer/detector assemblies are optimally placed the entrance of one 199 Hg atom into its analyzer/detector will almost certainly be accompanied by the entrance of the other atom into its assembly; the primary reason for the occasional failure of this coordination is the non-zero volume of the region in which the dissociation occurs. Since each of the 199 Hg atoms (with 80 electrons) is in the electronic ground state it will have zero electronic spin, and therefore the total spin F of each atom (which then equals its nuclear spin ½) will be F = ½. Given any choice of an axis, the only possible values of the magnetic quantum number F M relative to this axis are F M =½ and −½. The directions of this axis, θ 1 for atom 1 and θ 2 for atom 2, are the quantities to be used for the parameters a and b in the BCH Inequalities. The angles θ 1 and θ 2 are physically fixed by the directions of two left-circularly polarized 253.7 nm laser beams propagating in parallel planes each perpendicular to the plane in which the atoms 1 and 2 travel from the dissociation region. See Figure 2:

Figure 2 ( reprinted with permission )

This is a schematic of the experiment showing the direction of the mercury dimer beam, together with a pair of the dissociated atoms and their respective detection planes. The relative directions of the various laser beams are also shown.

The left-circular polarization ensures that if a photon is absorbed by atom 1 then F M of atom 1 will decrease by one, which is possible only if F M is initially ½, and likewise for absorption by atom 2. Thus each laser beam is selective and acts as an analyzer by taking only an F M =½ atom, 1 or 2 as the case may be, into a specific excited state. Detection is achieved in several steps. The first step is the impingement on atom 1 of a 197.3 nm laser beam, timed to arrive within the lifetime of the excited state produced by the 253.7 nm laser beam; absorption of a photon from this beam by 1 will cause ionization (and likewise an excited 2 is ionized). The second step is the detection of either the resulting ion or the associated photoelectron or both — the detection being a highly efficient process, since both of these products of ionization are charged. The first BCH Inequality predicts that the coincident detection rates and the single detection rates in this experiment satisfy

(47)  [ D (θ 1 , θ 2 ) + D (θ 1 , θ′ 2 ) + D (θ′ 1 , θ 2 ) − D (θ′ 1 , θ′ 2 )] / [ D 1 (θ 1 ) + D 2 (θ 2 )] ≤ 1.

Fry and Walther calculate that with proper choices of the four angles, large enough values of detector efficiencies, large enough probability of 1 entering the region of analysis, large enough probability of 2 entering its region of analysis conditional upon 1 doing so, and small enough probability of mistaken analysis (e.g., mistaking an F M = −½ for an F M = ½ because of rare processes) the quantum mechanical predictions will violate (47). If these predictions are fulfilled, no auxiliary assumption like “no enhancement” will be needed to disprove Inequality (43) experimentally. The detection loophole in the refutation of the family of Local Realistic Theories will thereby be closed. Fry and Walther express some warning, however, against excessive optimism about detecting a sufficiently large percentage of the ions and the electrons produced by the ionization of the Hg atoms, together with a low rate of errors due to background or noise counts: “the Hg partial pressure, as well as the partial pressure of all other residual gases, must be kept as low as possible. An ultra high vacuum of better than 10 -9 Torr is required and the detector must be cooled to liquid nitrogen temperatures to freeze out background Hg atoms… Equally important is the suppression of photoelectrons produced by scattered photons.” (Fry & Walther 1997, p. 67). If these desiderata are not achieved, it would be necessary to resort to BCH's second Inequality, which required the ”no enhancement” assumption.

The derivations of all the variants of Bell's Inequality depend upon the two Independence Conditions (8a,b) and (9a,b). Experimental data that disagree with a Bell's Inequality are not a refutation unless these Conditions are satisfied by the experimental arrangement. In the early tests of Bell's Inequalities it was plausible that these Conditions were satisfied just because the 1 and the 2 arms of experiment were spatially well separated in the laboratory frame of reference. This satisfaction, however, is a mere contingency not guaranteed by any law of physics, and hence it is physically possible that the setting of the analyzer of 1 and its detection or non-detection could influence the outcome of analysis and the detection or non-detection of 2, and conversely. This is the “communication loophole” in the early Bell tests. If the process of analysis and detection of 1 were an event with space-like separation from the event consisting of the analysis and detection of 2 then the satisfaction of the Independence Conditions would be a consequence of the Special Theory of Relativity, according to which no causal influences can propagate with a velocity greater than the velocity of light in vacuo. Several experiments of increasing sophistication between 1982 and the present have attempted to block the Communication Loophole in this way.

Aspect, Dalibard, and Roger (1982) published the results of an experiment in which the choices of the orientations of the analyzers of photons 1 and 2 were performed so rapidly that they were events with space-like separation. No physical modification was made of the analyzers themselves. Instead, switches consisting of vials of water in which standing waves were excited ultrasonically were placed in the paths of the photons 1 and 2. When the wave is switched off, the photon propagates in the zeroth order of diffraction to polarization analyzers respectively oriented at angles a and b , and when it is switched on the photons propagate in the first order of diffraction to polarization analyzers respectively oriented at angles a ′ and b ′. The complete choices of orientation require time intervals 6.7 ns and 13.37 ns respectively, much smaller than the 43 ns required for a signal to travel between the switches in obedience to Special Relativity Theory. Prima facie it is reasonable that the Independence Conditions are satisfied, and therefore that the coincidence counting rates agreeing with the quantum mechanical predictions constitute a refutation of Bell's Inequality and hence of the family of Local Realistic Theories. There are, however, several imperfections in the experiment. First of all, the choices of orientations of the analyzers are not random, but are governed by quasiperiodic establishment and removal of the standing acoustical waves in each switch. A scenario can be invented according to which the clever hidden variables of each analyzer can inductively infer the choice made by the switch controlling the other analyzer and adjust accordingly its decision to transmit or to block an incident photon. Also coincident count technology is employed for detecting joint transmission of 1 and 2 through their respective analyzers, and this technology establishes an electronic link which could influence detection rates. And because of the finite size of the apertures of the switches there is a spread of the angles of incidence about the Bragg angles, resulting in a loss of control of the directions of a non-negligible percentage of the outgoing photons.

The experiment of Tittel, Brendel, Zbinden, and Gisin (1998) did not directly address the communication loophole but threw some light indirectly on this question and also provided the most dramatic evidence so far concerning the maintenance of entanglement between particles of a pair that are well separated. Pairs of photons were generated in Geneva and transmitted via cables with very small probability per unit length of losing the photons to two analyzing stations in suburbs of Geneva, located 10.9 kilometers apart on a great circle. The counting rates agreed well with the predictions of Quantum Mechanics and violated one of Bell's Inequalities. No precautions were taken to ensure that the choices of orientations of the two analyzers were events with space-like separation. The great distance between the two analyzing stations makes it difficult to conceive a plausible scenario for a conspiracy that would violate Bell's Independence Conditions. Furthermore — and this is the feature which seems most to have captured the imagination of physicists — this experiment achieved much greater separation of the analyzers than ever before, thereby providing the best reply to date to a conjecture by Schrödinger (1935) that entanglement is a property that may dwindle with spatial separation.

The experiment that comes closest so far to closing the Communication Loophole is that of Weihs, Jennenwein, Simon, Weinfurter, and Zeilinger (1998). The pairs of systems used to test a Bell's Inequality are photon pairs in the entangled polarization state

(48)  |Ψ> = 1/√2 (| H > 1 | V > 2 − | V > 1 | H > 2 ),

where the ket | H > represents horizontal polarization and | V > represents vertical polarization. Each photon pair is produced from a photon of a laser beam by the down-conversion process in a nonlinear crystal. The momenta, and therefore the directions, of the daughter photons are strictly correlated, which ensures that a non-negligible proportion of the pairs jointly enter the apertures (very small) of two optical fibers, as was also achieved in the experiment of Tittel et al. The two stations to which the photon pairs are delivered are 400 m apart, a distance which light in vacuo traverses in 1.3 μs. Each photon emerging from an optical fiber enters a fixed two-channel polarizer (i.e., its exit channels are the ordinary ray and the extraordinary ray). Upstream from each polarizer is an electro-optic modulator, which causes a rotation of the polarization of a traversing photon by an angle proportional to the voltage applied to the modulator. Each modulator is controlled by amplification from a very rapid generator, which randomly causes one of two rotations of the polarization of the traversing photon. An essential feature of the experimental arrangement is that the generators applied to photons 1 and 2 are electronically independent. The rotations of the polarizations of 1 and 2 are effectively the same as randomly and rapidly rotating the polarizer entered by 1 between two possible orientations a and a ′ and the polarizer entered by 2 between two possible orientations b and b ′. The output from each of the two exit channels of each polarizer goes to a separate detector, and a “time tag” is attached to each detected photon by means of an atomic clock. Coincidence counting is done after all the detections are collected by comparing the time tags and retaining for the experimental statistics only those pairs whose tags are sufficiently close to each other to indicate a common origin in a single down-conversion process. Accidental coincidences will also enter, but these are calculated to be relatively infrequent. This procedure of coincidence counting eliminates the electronic connection between the detector of 1 and the detector of 2 while detection is taking place, which conceivably could cause an error-generating transfer of information between the two stations. The total time for all the electronic and optical processes in the path of each photon, including the random generator, the electro-optic modulator, and the detector, is conservatively calculated to be smaller than 100 ns, which is much less than the 1.3 μs required for a light signal between the two stations. With the choice made in Eq. (22) of the angles a , a ′, b , and b ′ and with imperfections in the detectors taken into account, the Quantum Mechanical prediction is

(49)  S ψ ≡ E ψ ( a , b  ) + E ψ ( a , b ′) + E ψ ( a ′, b  ) − E ψ ( a ′, b ′) = 2.82,

which is 0.82 greater than the upper limit allowed by the BCHSH Ineq. (16). The experimental result in the experiment of Weihs et al. is 2.73 +/- 0.02, in good agreement with the Quantum Mechanical prediction of Eq. (49), and it is 30 sd away from the upper limit of Ineq. (16). Aspect, who designed the first experimental test of a Bell Inequality with rapidly switched analyzers (Aspect, Dalibard, Roger 1982) appreciatively summarized the import of this result:

I suggest we take the point of view of an external observer, who collects the data from the two distant stations at the end of the experiment, and compares the two series of results. This is what the Innsbruck team has done. Looking at the data a posteriori, they found that the correlation immediately changed as soon as one of the polarizers was switched, without any delay allowing for signal propagation: this reflects quantum non-separability. (Aspect 1999)

The experiment of Weihs et al. does not completely block the detection loophole, and even if the experiment proposed by Fry and Walther is successfully completed, it will still be the case that the detection loophole and the communication loophole will have been blocked in two different experiments. It is therefore conceivable — though with difficulty, in the subjective judgment of the present writer — that both experiments are erroneous, because Nature took advantage of a separate loophole in each case. For this reason Fry and Walther suggest that their experiment using dissociated mercury dimers can in principle be refined by using electro-optic modulators (EOM), so as to block both loopholes: “Specifically, the EOM together with a beam splitting polarizer can, in a couple of nanoseconds, change the propagation direction of the excitation laser beam and hence the component of nuclear spin angular momentum being observed. A separation between our detectors of approximately 12 m will be necessary in order to close the locality loophole” (Fry & Walther 2002) [See Fig. 2 and also note that “locality loophole” is their term for the communication loophole.]

In the face of the spectacular experimental achievement of Weihs et al. and the anticipated result of the experiment of Fry and Walther there is little that a determined advocate of local realistic theories can say except that, despite the spacelike separation of the analysis-detection events involving particles 1 and 2, the backward light-cones of these two events overlap, and it is conceivable that some controlling factor in the overlap region is responsible for a conspiracy affecting their outcomes. There is so little physical detail in this supposition that a discussion of it is best delayed until a methodological discussion in Section 7 .

6. Some Variants of Bell's Theorem

This section will discuss in some detail two variants of Bell's Theorem which depart in some respect from the conceptual framework presented in Section 2 . Both are less general than the version in Section 2 , because they apply only to a deterministic local realistic theory — that is a theory in which a complete state m assigns only probabilities 1 or 0 (‘yes’ or ‘no’) to the outcomes of the experimental tests performed on the systems of interest. By contrast, the Local Realistic Theories studied in Section 2 are allowed to be stochastic, in the sense that a complete state can assign other probabilities between 0 and 1 to the possible outcomes. At the end of the section two other variants will be mentioned briefly but not summarized in detail.

The first variant is due independently to Kochen and Specker (1967), Heywood and Redhead (1983), and Stairs (1983). Its ensemble of interest consists pairs of spin-1 particles in the entangled state

(50)  |Φ> = 1/√3 [ | z ,1> 1 | z ,−1> 2 − | z ,0> 1 | z ,0> 2 + | z ,−1> 1 | z ,1> 2 ],

where | z , i > 1 , with i = −1 or 0 or 1 is the spin state of particle 1 with component of spin i along the axis z , and | z , i > 2 has an analogous meaning for particle 2. If x , y , z is a triple of orthogonal axes in 3-space then the components s x , s y , s z of the spin operator along these axes do not pairwise commute; but it is a peculiarity of the spin-1 system that the squares of these operators — s x 2 , s y 2 , s z 2 — do commute, and therefore, in view of the considerations of Section 1 , any two of them can constitute a context in the measurement of the third. If the operator of interest is s z 2 , the axes x and y can be any pair of orthogonal axes in the plane perpendicular to z , thus offering an infinite family of contexts for the measurement of s z 2 . As noted in Section 1 Bell exhibited the possibility of a contextual hidden variables theory for a quantum system whose Hilbert space has dimension 3 or greater even though the Bell-Kochen-Specker theorem showed the impossibility of a non-contextual hidden variables theory for such a system. The strategy of Kochen and of Heywood-Redhead is to use the entangled state of Eq. (50) to predict the outcome of measuring s z 2 for particle 2 (for any choice of z  ) by measuring its counterpart on particle 1. A specific complete state m would determine whether s z 2 of 1, measured together with a context in 1, is 0 or 1. Agreement with the quantum mechanical prediction of the entangled state of Eq. (50) implies that s z 2 of 2 has the same value 0 or 1. But if the Locality Conditions (8a,b) and (9a,b) are assumed, then the result of measuring s z 2 on 2 must be independent of the remote context, that is, independent of the choice of s x 2 and s y 2 of 1, hence of 2 because of correlation, for any pair of orthogonal directions x and y in the plane perpendicular to z . It follows that the Local Realistic Theory which supplies the complete state m is not contextual after all, but maps the set of operators s z 2 of 2, for any direction z , noncontextually into the pair of values (0, 1). But that is impossible in view of the Bell-Kochen-Specker theorem. The conclusion is that no deterministic Local Realistic Theory is consistent with the Quantum Mechanical predictions of the entangled state (50). An alternative proof is thus provided for an important special case of Bell's theorem, which was the case dealt with in Bell's pioneering paper of 1964: that no deterministic local realistic theory can agree with all the predictions of quantum mechanics. An objection may be raised that s z 2 of 1 is in fact measured together with only a single context — e.g., s x 2 and s y 2 — while other contexts are not measured, and “unperformed experiments have no results” (a famous remark of Peres 1978). It may be that this remark is a correct epitome of the Copenhagen interpretation of quantum mechanics, but it certainly is not a valid statement in a deterministic version of a Local Realistic interpretation of Quantum Mechanics, because a deterministic complete state is just what is needed as the ground for a valid counterfactual conditional. We have good evidence to this effect in classical physics: for example, the charge of a particle, which is a quantity inferred from the actual acceleration of the particle when it is subjected to an actual electric field, provides in conjunction with a well-confirmed force law the basis for a counterfactual proposition about the acceleration of the particle if it were subjected to an electric field different from the actual one.

A simpler proof of Bell's Theorem, also relying upon counterfactual reasoning and based upon a deterministic local realistic theory, is that of Hardy (1993), here presented in Laloë's (2001) formulation. Consider an ensemble of pairs 1 and 2 of spin-½ particles, the spin of 1 measured along directions in the xz -plane making angles a =θ/2 and a ′=0 with the z -axis, and angles b and b ′ having analogous significance for 2. The quantum states for particle 1 with spins +½ and −½ relative to direction a ′ are respectively | a ′,+> 1 and | a ′,−> 1 , and relative to direction a are respectively

(51a)  | a ,+> 1 = cosθ| a ′,+> 1 + sinθ| a ′,−> 1

(51b)  | a ,−> 1 = -sinθ| a ′,+> 1 + cosθ| a ′,−> 1 ;

the spin states for 2 are analogous. The ensemble of interest is prepared in the entangled quantum state

(52)  |Ψ> = -cosθ| a ′,+> 1 | b ′,−> 2 − cosθ| a ′,−> 1 | b ′,+> 2 + sinθ| a ′,+> 1 | b ′,+> 2

(unnormalized, because normalization is not needed for the following argument). Then for the specified a , a ′, b , and b ′ the following quantum mechanical predictions hold:

(53)  <Ψ| a ,+> 1 | b ′,+> 2 = 0;

(54)  <Ψ| a ′,+> 1 | b ,+> 2 = 0 ;

(55)  <Ψ| a ′,−> 1 | b ′,−> 2 = 0;

and for almost all values of the θ of Eq. (52)

(56)  <Ψ| a ,+> 1 | b ,+> 2 ≠ 0 ,

with the maximum occurring around θ = 9 o . Inequality (56) asserts that for the specified angles there is a non-empty subensemble E′ of pairs for which the results for a spin measurement along a for 1 and along b for 2 are both +. Eq. (53) implies the counterfactual proposition that if the spin of a 2 in E′ had been measured along b ′ then with certainty the result would have been −; and likewise Eq. (54) implies the counterfactual proposition that if the spin of a 1 in E′ had been measured along a ′ then with certainty the result would have been −. It is in this step that counterfactual reasoning is used in the argument, and, as in the argument of Kochen-Heywood-Redhead-Stairs in the previous paragraph, the reasoning is based upon the deterministic Local Realistic Theory. Since the subensemble E′ is non-empty, we have reached a contradiction with Eq. (55), which asserts that if the spin of 1 is measured along a ′ and that of 2 is measured along b ′ then it is impossible that both results are −. The incompatibility of a deterministic Local Realistic Theory with Quantum Mechanics is thereby demonstrated.

An attempt was made by Stapp (1997) to demonstrate a strengthened version of Bell's theorem which dispenses with the conceptual framework of a Local Realistic Theory and to use instead the logic of counterfactual conditionals. His intricate argument has been the subject of a criticism by Shimony and Stein (2001, 2003), who are critical of certain counterfactual conditionals that are asserted by Stapp by means of a “possible worlds” analysis without a grounding on a deterministic Local Realistic Theory, and a response by Stapp (2001) himself, who defends his argument with some modifications.

The three variants of Bell's Theorem considered so far in this section concern ensembles of pairs of particles. An entirely new domain of variants is opened by studying ensembles of n -tuples of particles with n ≥3. The prototype of this kind of theorem was demonstrated by Greenberger, Horne, and Zeilinger (1989) (GHZ) for n =4 and modified to n =3 by Mermin (1990) and by Greenberger, Horne, Shimony, and Zeilinger (1990) (GHSZ). In the theorem of GHZ an entangled quantum state was written for four spin ½ particles and the expectation value of the product of certain binary measurements performed on the individual particles was calculated. They then showed that the attempt to duplicate this expectation value subject to the constraints of a Local Realistic Theory produces a contradiction. A similar result was obtained by Mermin for a state of 3 spin-½ particles and by GHSZ for a state of 3 photons entangled in direction. Because of the length of these arguments and limitations of space in the present article the details will not be summarized here. Furthermore, for the philosophically crucial purpose of demonstrating experimentally the validity of Quantum Mechanical predictions and the violation of the corresponding predictions of Local Realistic Theories the examples using pairs of particles in Section 3 are more promising than n -tuple experiments with n ≥3. In particular, it is evident that the detection loophole is more difficult to block in an experiment performed with n -tuples of particles, n ≥3 , than in an experiment using pairs, because the net efficiency of detecting n -tuples is proportional to the product of the efficiencies of the detectors of the individual particles.

Two different classes of philosophical questions are raised by reflection upon the theoretical and experimental investigations concerning Bell's Theorem. Questions of one class are logical and methodological: whether one can legitimately infer from these investigations that quantum mechanics is non-local, and whether the experimental data definitively prove that Bell's Inequalities are violated. Questions of the other class are metaphysical: upon assumption that the logical and methodological questions are answered positively, what conclusions can be drawn about the structure and constitution of the physical world, in particular is nature non-local despite the remarkable success of relativity theory?

A logical question has been raised by Fine. In a paper (Fine 1999), which analyzes the construction of Hardy discussed in Section 6 , Fine concludes with a philosophical thesis: “That means that the Hardy theorem, like other variants on Bell, is not a ‘proof of nonlocality’. It is a proof that locality cannot be married to the assignment of determinate values in the recommended way. That is interesting and significant. It is not, however, a demonstration that quantum mechanics is nonlocal, much less (as some proclaim) that nature is.” Fine's analysis of Hardy's construction relies upon his earlier paper (Fine 1982b) which contains the following theorem (which is a combination of his Theorem 4, p. 1309, and footnote 5 on p. 1310):

For a correlation experiment with observables A 1 , A 2 , B 1 , B 2 and with exactly the four pairs A i , B j ( i = 1,2; j = 1,2) commuting, the following statements are equivalent: (I) the Bell/CH inequalities hold for the single and double probabilities of the experiment; (II) there is a joint distribution P ( A 1 , A 2 , B 1 , B 2 ) compatible with the observed single and double distributions; (III) there is a deterministic hidden variables theory for A 1 , A 2 , B 1 , B 2 returning the observed single and double distributions; (IV) there is a well-defined joint distribution (for the noncommuting pairs B 1 , B 2 ) and joint distributions P ( A 1 , B 1 , B 2 ),each of the latter compatible with B 1 , B 2 and with the observed single and double distributions; (V) there exists a factorizable (so-called “local”) stochastic hidden variables theory for A 1 , A 2 , B 1 , B 2 returning the observed single and double distributions. [ 3 ]

Proposition (I) is the one of the five propositions in this theorem which is amenable to direct experimental confirmation or disconfirmation. I shall accept for the present the experimental disconfirmation of some of these Inequalities, leaving a consideration of methodological doubts about this disconfirmation for discussion below. With this proviso it follows that each of the propositions (II), (III), (IV), and (V) is disconfirmed by modus tollens. The disconfirmation of (V) logically implies the falsity of the conjunction of all the premisses from which the Bell/CH (called “BCH” in Section 4 ) Inequalities are inferred: namely, the assumptions (5), (6), and (7) in Section 2 about the existence of well-defined single and double probabilities, and the Independence Conditions (8a,b) and (9a,b), respectively called “Remote Outcome Independence” and “Remote Context Independence.” Usually the assumptions (5), (6), and (7) are not doubted, for two reasons: first, they are implicit in the pervasive assumption in hidden-variables investigations that the phenomenological assertions of quantum mechanics are correct — an assumption which even permits us to use the concept of a “complete state”, denoted by m , which is the quantum state itself if there are no hidden variables, or the complete specification of the hidden variables if such entities exist; second, there is overwhelming experimental confirmation of these assumptions by the practical success of quantum mechanics, not just in experimentation regarding hidden-variables hypotheses. (In spite of these weighty considerations there is one important program which attempts to weaken or replace assumptions (5), (6), and (7), namely that of Stapp, briefly discussed in Section 6 .) Consequently, the falsity of the conjunction of the premisses from which the BCH Inequality is derived implies the falsity of one or both of the Independence Conditions (8a,b) and (9a,b). Since the failure of either of these Conditions is prima facie in contradiction with relativistic locality, it is not important for the present concern to investigate which of the two Independence Conditions is weaker — a question that will be taken up later in Section 7 . The conclusion at this stage in the argument, when propositions (II), (III), and (IV) of Fine's theorem have been neglected, is that the experimental disconfirmation of the BCH Inequalities does imply the occurrence of non-locality in natural phenomena and since the quantum mechanical analysis of pairs of systems in entangled states anticipates non-local phenomena, quantum mechanics itself is a a non-local theory. This pair of conclusions is what Fine claims, in the passage from (Fine 1999) quoted above, has not been demonstrated. What he allows, in virtue of his theorem is the weaker conclusion that “locality cannot be married to the assignment of determinate values in the recommended way.”

But is this retrenchment to a weaker conclusion logically justified? The strong conclusion that quantum mechanics and nature are non-local has been derived from one part of the theorem — that (V) implies (I) — together with some auxiliary analysis of premisses (5), (6), and (7). The other parts of the theorem — the equivalences of (II), (III), (IV) to each other and to (I) and (V) — provide information supplementary to that provided by the equivalence of (I) and (V), but as a matter of logic do not diminish the information given by the last equivalence.

The last resort of a dedicated adherent of local realistic theories, influenced perhaps by Einstein's advocacy of this point of view, is to conjecture that apparent violations of locality are the result of conspiracy plotted in the overlap of the backward light cones of the analysis-detection events in the 1 and 2 arms of the experiment. These backward light cones always do overlap in the Einstein-Minkowski space-time of Special Relativity Theory (SRT) — a framework which can accommodate infinitely many processes besides the standard ones of relativistic field theory. Elimination of any finite set of concrete scenarios to account for the conspiracy leaves infinitely many more as unexamined and indeed unarticulated possibilities. What attitude should a reasonable scientist take towards these infinite riches of possible scenarios? We should certainly be warned by the power of Hume's skepticism concerning induction not to expect a solution that would be as convincing as a deductive demonstration and not to expect that the inductive support of induction itself can fill the gap formed by the shortcoming of a deductive justification of induction (Hume 1748, Sect. 4). One solution to this problem is a Bayesian strategy that attempts to navigate between dogmatism and excessive skepticism (Shimony 1993, Shimony 1994). To avoid the latter one should keep the mind open to a concrete and testable proposal regarding the mechanism of the suspected conspiracy in the overlap of the backward light cones, giving such a proposal a high enough prior probability to allow the possibility that its posterior probability after testing will warrant acceptance. To avoid the former one should not give the broad and unspecific proposal that a conspiracy exists such high prior probability that the concrete hypothesis of the correctness of Quantum Mechanics is debarred effectively from acquiring a sufficiently high posterior probability to warrant acceptance. This strategy actually is implicit in ordinary scientific method. It does not guarantee that in any investigation the scientific method is sure to find a good approximation to the truth, but it is a procedure for following the great methodological maxim: “Do not block the way of inquiry” (Peirce 1931). [ 4 ]

A second solution, which can be used in tandem with the first, is to pursue theoretical understanding of a baffling conceptual problem that at present confronts us: that the prima facie nonlocality of Quantum Mechanics will remain a permanent part of our physical world view, in spite of its apparent tension with Relativistic locality. This solution opens the second type of philosophical questions mentioned in the initial paragraph of the present section, that is, metaphysical questions about the structure and constitution of the physical world. Among the proposals for a solution of the second kind are the following.

Do we then have to fall back on ‘no signaling faster than light’ as the expression of the fundamental causal structure of contemporary theoretical physics? That is hard for me to accept. For one thing we have lost the idea that correlations can be explained, or at least this idea awaits reformulation. More importantly, the ‘no signaling …’ notion rests on concepts that are desperately vague, or vaguely applicable. The assertion that ‘we cannot signal faster than light’ immediately provokes the question: Who do we think we are? We who make ‘measurements,’ we who can manipulate ‘external fields,’ we who can ‘signal’ at all, even if not faster than light. Do we include chemists, or only physicists, plants, or only animals, pocket calculators, or only mainframe computers? (Bell 1990, Sec. 6.12)

There may indeed be “peaceful coexistence” between Quantum nonlocality and Relativistic locality, but it may have less to do with signaling than with the ontology of the quantum state. Heisenberg's view of the mode of reality of the quantum state was briefly mentioned in Section 2 — that it is potentiality as contrasted with actuality . This distinction is successful in making a number of features of quantum mechanics intuitively plausible — indefiniteness of properties, complementarity, indeterminacy of measurement outcomes, and objective probability. But now something can be added, at least as a conjecture: that the domain governed by Relativistic locality is the domain of actuality, while potentialities have careers in space-time (if that word is appropriate) which modify and even violate the restrictions that space-time structure imposes upon actual events. The peculiar kind of causality exhibited when measurements at stations with space-like separation are correlated is a symptom of the slipperiness of the space-time behavior of potentialities. This is the point of view tentatively espoused by the present writer, but admittedly without full understanding. What is crucially missing is a rational account of the relation between potentialities and actualities — just how the wave function probabilistically controls the occurrence of outcomes. In other words, a real understanding of the position tentatively espoused depends upon a solution to another great problem in the foundations of quantum mechanics − the problem of reduction of the wave packet.

The quantum state is exactly that representation of our knowledge of the complete situation which enables the maximal set of (probabilistic) predictions of any possible future observation. What comes new in quantum mechanics is that, instead of just listing the various experimental possibilities with the individual probabilities, we have to represent our knowledge of the situation by the quantum state using complex amplitudes. If we accept that the quantum state is no more than a representation of the information we have, then the spontaneous change of the state upon observation, the so-called collapse or reduction of the wave packet, is just a very natural consequence of the fact that, upon observation, our information changes and therefore we have to change our representation of the information, that is, the quantum state. (1999, p. S291).

This point of view is very successful at accounting for the arbitrarily fast connection between the outcomes of correlated measurements, but it scants the objective features of the quantum state. Especially it scants the fact that the quantum state probabilistically controls the occurrence of actual events.

• A radical idea concerning the structure and constitution of the physical world, which would throw new light upon quantum nonlocality, is the conjecture of Heller and Sasin (1999) about the nature of space-time in the very small, specifically at distances below the Planck length (about 10 -33 cm). Quantum uncertainties in this domain have the consequence of making ill-defined the metric structure of General Relativity Theory. As a result, according to them, basic geometric concepts like point and neighborhood are ill-defined, and non-locality is pervasive rather than exceptional as in atomic, nuclear, and elementary particle physics. Our ordinary physics, at the level of elementary particles and above, is (in principle, though the details are obscure) recoverable as the correspondence limit of the physics below the Planck length. What is most relevant to Bell's Theorem is that the non-locality which it makes explicit in Quantum Mechanics is a small indication of pervasive ultramicroscopic nonlocality. If this conjecture is taken seriously, then the baffling tension between Quantum nonlocality and Relativistic locality is a clue to physics in the small. Regrettably we not longer have John Bell, with his incomparable analytic powers, to comment on this radical proposal.

Although the main result of Bell [1964] is his theorem demonstrating the the impossibility of recovering the statistical predictions of quantum mechanics with a local realistic theory, Section 3 of this paper concludes with the construction of a nonlocal model — violating Remote Context Independence but not Remote Outcome Independence-- which does recover the statistical predictions of a particular entangled quantum state. Recently several investigators have investigated the resources of nonlocal realistic theories and have demonstrated that certain important subclasses of nonlocal theories are also incompatible with certain statistical predictions of quantum mechanics. Particularly interesting are Leggett [2003], Groeblacher et al . [2007], and Branciard et al . [2008]. A conceptually important open problem is to demonstrate necessary and sufficient conditions for a specified class of nonlocal realistic theories to recover the statistical predictions of an arbitrary quantum mechanical entangled state by an appropriate choice of the space of hidden variables and of the probability distribution over this space.

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quantum mechanics | quantum mechanics: Bohmian mechanics | quantum mechanics: Kochen-Specker theorem | quantum theory: the Einstein-Podolsky-Rosen argument in

Acknowledgments

Valuable conversations with John Clauser and Edward Fry are gratefully acknowledged. Springer Verlag and Alain Aspect have kindly given permission to reproduce Figure 9.1 on p. 121 of Aspect's article, “Bell's Theorem: the Naïve View of an Experimentalist,” pp. 119–153 in Quantum [Un]speakables , R.A. Bertlmann and A. Zeilinger (eds.), Berlin-Heidelberg-New York: Springer Verlag, 2002; this Figure was used as Figure 1 in the present article. Kluwer Academic Publishers and Edward Fry have kindly given permission to reproduce Figure 2 on p. 66 of E.S. Fry's and T. Walther's article “A Bell Inequality Experiment Based on Molecular Dissociation — Extension of the Lo-Shimony Proposal to 199 Hg (Nuclear Spin ½) Dimers,” pp. 61–71 in Experimental Metaphysics , R.S. Cohen, M. Horne, and J. Stachel (eds.), Dordrecht: Kluwer, 1997; this Figure was used as Figure 2 in the present article. Dr. Caroline Thompson has kindly brought to my attention errors in Eqs. (13a,b,c,d), (43), and (46).

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Bell's theorem for dummies, how does it work?

I've been reading up on theoretical physics for a few years now and I feel like I am starting to get an understanding of particle physics, at least as much as you can from Wikipedia pages. One thing I have tried to understand but fails to make sense to me is Bell's Theorem . I understand that the premise of the ERP paper was that the "wave form collapse" couldn't work because it would require the two particles which made up the convoluted wave form to communicate instantly, violating the information speed limit. Instead they suggested that it was due to hidden variables (ie the values are already set, whether they have been measured or not).

My question is can any one explain to me how Bell's experiment works and how it disproves this in terms that don't require an in-depth understanding of the math behind quantum mechanics?

My current understanding of the experiment is you have two people who are reading a quantum value of entangle quantum particles (for my understanding lets say the spin state of a positron-electron pair produced by a pair production event). For each particle pair the two readers measure the spin at a randomly chosen angle.

Here is where I need clarification: if I understand it correctly, local realism hypothesis states that when measured on the same axis the spin states should always be opposite (.5 + -.5 =0, ie conservation) when measure on opposite axis the spin states should always be the same ( .5 - .5 = 0 ) and when measured 90 degrees apart, the values are totally random. This part I get. I believe these results are predicted to be the same by both local realism and quantum mechanics. The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?

What I would like to have explained is the following:

What are the predictions made by quantum mechanics?

What are the predictions made by local realism?

How do they differ?

How is entanglement different from conservation?

Any corrections in regard to my explanation above?

• quantum-mechanics
• quantum-entanglement
• bells-inequality
• epr-experiment

• 1 $\begingroup$ Sorry, could you try to clarify your fourth question? I don't quite understand what you're asking there... $\endgroup$ –  Danu Commented May 24, 2014 at 17:11
• $\begingroup$ I guess this is the root question in a way. Conservations says when ever to particles interact the sum of their quantum values must equal that of the progenitors. Ie pair production makes a pair of particles whose charge, spin, momentum etc all are equal to the particle (photon) which created it. There for if you know the states of the photon and the states of one of the particles, then you know the state of the other particle. I guess this is realism, and if you answer the other questions, you'll answer this question. $\endgroup$ –  jeffpkamp Commented May 24, 2014 at 17:32
• 1 $\begingroup$ Perhaps an explanation via sets comparing them would help? youtube.com/watch?v=qd-tKr0LJTM $\endgroup$ –  user6972 Commented May 27, 2014 at 17:36
• 3 $\begingroup$ I like this article by David Mermin, "Quantum mysteries for anybody", web.pdx.edu/~pmoeck/pdf/Mermin%20short.pdf , and for a simple proof, I don't think you can do better than this arxiv.org/abs/1212.5214 $\endgroup$ –  innisfree Commented Jan 27, 2015 at 15:31
• $\begingroup$ @innisfree if I understand correctly, the key concept here is that Bell devised an experiment which would yield different statistical results for locality and for QM equations. The expected results for locality are straightforward to understand. It seems the difficulty lies in visualizating / understanding the meaning of the results predicted by QM and perhaps even just understanding QM maths. Is that an accurate assessment? $\endgroup$ –  DPM Commented Apr 11, 2020 at 23:49

6 Answers 6

Bell's theorem shows that standard QM is inconsistent with local realism . Local realism is a very general principle that was not originally thought to make any testable physical predictions. A major part of Bell's achievement was showing that Bell's inequality is implied by local realism , while standard QM predictions violate it . Experiments like Aspect's have since shown that Bell's inequalities are violated in reality, refuting local realism, in a way that is consistent with standard QM.

I think your issue is with the definition of local realism:

when measured on the same axis the spin states should always be opposite (.5 + -.5 = 0, ie conservation) when measured on the opposite axis the spin states should always be the same (.5 - .5 = 0) and when measured 90 degrees apart, the values are totally random.

This is just what standard QM predicts for entangled particles.

Local realism states that what happens at any point can only be directly affected by the state in its immediate neighbourhood, any long range effects must be mediated by particles or field disturbances travelling at (sub)luminal velocities, and that all behaviour is deterministic.

If entangled particles are far enough apart that one can perform measurements on both of them in a way that ensures the measurement events are separated by a space-like interval then local realism would require the particles to carry enough hidden variables to predetermine the outcome of each possible measurement, since any effect from one measurements would not have time to propagate to the other measurement to enforce the correlated observations.

Local realism and Bell's inequalities are not violated when only measurements separated by integer multiples of 90 degrees like in your description are considered. The discrepancy between QM and local realism only appears when oblique angles are considered, reaching a maximum when the angle between the measurements is 45 degrees (plus some multiple of 90 degrees), when the correlation between the measurements becomes $\sqrt{2}$ greater than allowed by Bell's inequality and therefore by local realism.

Spin conservation is really a separate issue. It just says that if the total spin of an isolated system was $x$ at some point in the past then it will always be $x$ and vice versa. Entanglemnt provides a way of satisfying conservation laws without assigning definite values of the conserved quantities to the individual components.

Bell's theorem is really about local realism and not really about QM. Experimental results could in principle violate Bell's inequality but not agree with QM predictions either. This would still rule out local realism and all theories satisfying it. The fact that QM does predict correlations higher than allowed by Bell's inequality and experimental results do agree with those predictions is kind of incidental.

• 2 $\begingroup$ I already understood most of what you described. The part that I don't understand is how the tests for bells inequalities work and what the results mean. The idea behind entanglement is that measurement of one particle alters its partner in some detectable way when measured at 45 degree angle ways, whereas L-R says measuring one doesn't affect the other because the values are already set. How do Bell tests prove the QM scenario? Are +45 angles more or less correlated than expected? Has anyone done a blind experiment where an eavesdropper could secretly randomly sample and show an effect? $\endgroup$ –  jeffpkamp Commented May 27, 2014 at 14:00
• $\begingroup$ The essence of what Bell experiments really do is make measurements on a large number of entangled particle pairs and observe the correlation over the entire set. If it is higher than the Bell inequality then that contradicts LR. There is not observable effect of one particle on another when measuring an entangled particle. It is just in the statistics. The real meat is in Bell's proof that LR implies Bell's inequality. This means that experiments that contradict Bell's inequality also contradict LR. The correlation predicted by QM is higher than allowed by LR via Bell's inequality. $\endgroup$ –  Daniel Mahler Commented May 27, 2014 at 17:03
• $\begingroup$ You assumed that hidden variables should be integer, while the hidden variable can take real numbers as well. In the case of Bell's inequality , local realism and quantum mechanics does not show any difference if we consider hidden variables as real numeric variables. $\endgroup$ –  Alberto Commented Sep 13, 2016 at 20:31
• $\begingroup$ If the experiment was designed to disprove LR, then 1) It wasn't a thought experiment, right? and 2) Why would Bell suspect that the inequality would be violated in the real world? Was it some consequence derived from QM for such experiment, like, I don't know, that the spins change a little with each measurement? $\endgroup$ –  DPM Commented Apr 11, 2020 at 22:43

To understand Bell's theorem it is not at all necessary to know anything about quantum mechanics. Essentially, it is sufficient if you believe that quantum theory predicts that it is violated even if the two measurements are space-like separated so that getting information what is measured at the other place is forbidden even by relativity.

http://ilja-schmelzer.de/realism/game.php gives a simple explanation how Bell's theorem works.

Bell proves first that, once both measure the same direction they get 100% correlation, but there cannot be information what has been measured on the other side, all measurement results should be predefined. Then he chooses three angles 0, 120 and 240 degrees. Assume now both measure different angles. Then we know two of the three values, all predefined, all + or -. Once out of three values + or - there is at least one pair equal, the probability of getting equal results should be at least 1/3.

Quantum theory predicts only 1/4 of getting equal results.

The straightforward solution, the one which is realized in the existing hidden variable theories like the de Broglie-Bohm interpretation, is that one of the hidden variables is a hidden preferred frame, and that the hidden variables can send information faster than light. But a hidden preferred frame, even if contradicted by nothing, is anathema in modern physics, and people prefer to reject realism, causality, logic and everything else, and fall into complete mysticism, only to avoid a preferred frame.

• $\begingroup$ The problem is achieving perfect correlation. David Mermin brings up an interesting point in his book “Boojums All The Way Through”. Chapter 12 gives an excellent description of the experiment and part four shows how incredibly difficult it is to achieve perfect correlations. Usually the data comes only from runs in which both detectors actually flash or flash close enough together to assume their correlated. In the end this cherry picking changes the outcome. It’s possible that Un-skewed results could agree with quantum mechanics predictions. $\endgroup$ –  Bill Alsept Commented Nov 13, 2017 at 17:32
• $\begingroup$ To avoid this purely practical problem, there are the CHSH inequalities, which do not have this necessity. Slightly more complicated to understand the trick mathematically, so Bell's theorem in the original form remains easier to understand, but for practical tests one does not use it. $\endgroup$ –  Schmelzer Commented Nov 14, 2017 at 18:34
• 1 $\begingroup$ hyperlink ilja-schmelzer.de/realism/game.php are no longer working. Could be temporary. $\endgroup$ –  Duke William Commented May 30, 2023 at 9:53

My understanding is that the measurement at 45° matches the measurements at 0° and 90° more than it should (assuming local hidden variables) given how often 0° and 90° match.

Think of two detectors that move between 0°, 45°, and 90°, so that you get the 90° measurement when one is at 0° and the other at 90° and the 45° measurement when one is at 45° and the other at either 90° or 0°. When measuring 45° and one of the other two angles, you get a match 85% of the time. So 90° matches 45° 85% of the time, and 0° matches 45° 85% of the time – how often must 90° and 0° match? At least 70% of the time – 0°, 45°, and 90° would all match 70% of the time, and for the other 30%, half the time 45° would match with 0° and half the time it would match with 90°. 45° would match either angle 85% of the time - 70% when all three angles match, plus 15% when 45° matches one but not the other.

But when 90° and 0° are measured, they only match 50% of the time. What’s the most that 45° can equally match the other two? 50% of the time all three match, then the other 50% of the time when 90° and 0° do not match, 45° can only match one or the other. If it matches one half the time and the other the other half of the time, the highest percentage you can get is 75%. 50% for when all three match, then 25% of the time matching 90° and not 0° and 25% of the time matching 0° and not 90°.

So to answer your questions:

• What we actually see - 45° measurements matching 85% of the time, 90° measurements matching 50% of the time. This suggest that the angle of measurement of one particle has a correlation with the angle results of the measurement on the other particle.
• Two separate things. If just looking at the results of 45° measurements, it says that 90° measurements must match at least 70% of the time (70% of the time when 0°, 45°, 90° all match plus 15% each for 0° and 90° when 45° matches one and not the other) . However, if looking at 0° and 90°, then it says that 45° can’t match the other two more than 75% of the time (50% when all three match plus 25% for each angle when 45° matches them and not the other).
• Quantum predictions say that there can be a correlation between the angle of measurement of one particle and the result of the measurement on the other - even when there isn’t enough time between the final setting of the angle of measurement for one particle and the measuring of the other for light to travel between the two locations.
• The correlation between the particles are connected to actions done to one of the particles.
• I’d only argue with the wording of “completely random” for the 90° angle

I found this page useful for understanding the general concepts involved.

I will attempt to answer questions 1-3 as best I can. The others, I defer to the other excellent answers here provided.

Before I start: "The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?" -- correct.

2. What are the predictions made by local realism ?

I think that this is really the crux of the problem, regardless of what the predictions made by quantum mechanics are. This is because Bell's Inequality does not state a prediction of QM -- it states a prediction of local realism (or of other sets of closely related philosophies, such as locality + counterfactual definiteness ) -- and there is lots of evidence that the prediction of local realism made by Bells' Inequality does not hold . Thus, what QM predicts is only relevant if you are interested in one of its many interpretations to replace local realism. Of course, via these same experiments, the results tend to match the predictions of QM, so they also provide evidence for the QM equations, but I think that this not the main purpose of Bell's Inequality.

Bell's Inequality is a very abstract statement designed to cover any local realism theory. So, since intuition is what we are after, allow me to propose a particular local realism theory, which the experiments for Bell's Inequality will therefore provide equally good evidence against:

Hypothesis 1: The spin of a particle is governed by a hidden variable $\theta \in [-\pi,\pi)$ . Denote $\theta_\phi$ to be the outcome of measuring a particle's spin when our measuring equipment is calibrated at an angle $\phi$ (so for all $\phi$ , $\theta_\phi = 1$ or $\theta_\phi = -1$ ). In other words, even though we always measure spin to be up or down, there is some "hidden variable", $\theta$ , that is a continuous-valued spin that is the "real variable" which is the "true" spin, we having only a poor window with which to view it, namely, $\theta_\phi$ . For the sake of concreteness, we hypothesize the following behind-the-scenes mechanism for our measuring equipment:

$$\theta_\phi = \text{sgn}(\sin(\theta-\phi))$$

(this is a square-wave by the way... in a sense, you just round the angle you are measuring relative to the angle of your equipment. e.g. if $\phi=0$ and $\theta$ is negative, you get "down" and if the $\theta$ is positive you get "up")

Note that Hypothesis 1 gives us a mechanism for local realism since a particle has a definite spin (realism) given by $\theta$ . Also, an explanation for correlations between measurements when we study pairs of particles at a particular "angle" can now be explained by a local property, $\theta$ .

The next few paragraphs operate under the assumption of Hypothesis 1.

Now, per the usual example , let's generate sets of pairs of particles with opposite orientations. Let's just focus on a handful of pairs that we managed to generate, and let's pretend that we can peek under-the-covers to see: $\{(\theta_1, \theta_2)\} = \{(\pi/4,5\pi/4), (4\pi/3,\pi/3), (0.001, 0.001+\pi)\}$ . We want to think about what happens when we measure these particles. Calibrate detector A at $\phi=0$ and detector B at $\phi=\pi$ . If we split up the pairs, and send $[\pi/4,4\pi/3,0.001]$ to detector A, and $[5\pi/4,\pi/3,0.001 + \pi]$ to detector B, what do we expect to get? Plugging stuff into the above formula, we expect to get $[1, -1, 1]$ at detector A and $[1, -1, 1]$ at detector B. Play with the calibrations at detector A and B and re-plug stuff into the above equation. Note that no matter what you set them at, as long as they are opposite ( $\phi_\text{A} - \phi_\text{B} = \pi$ ), then we get identical results at detectors A and B (although perhaps not the exact sequence $[1, -1, 1]$ , depending on the calibration).

Now, notice that if we change the calibration of only A by a very small angle $\phi_\text{A} = 0.002$ , then the value of the third particle in our list $a_3$ at detector A will flip. The corresponding measurement at B should not, because we have not altered its calibration, and $\theta$ for $b_3$ remains the same. In other words, if we don't change the calibration of B $\phi_B$ , and we don't change any of the $\theta$ of our particles, then it is of no consequence what is happening over at A, whether researchers are measuring particles, or whether they have all gone to have a beer, what we measure at B is totally unaffected and has only to do with the calibration and particles at B. This statement is a necessary condition for local realism to hold. If, somehow, the corresponding measurement at B changes depending on whether an observation occurred at A, then either it communicated with its pair particle over at A (to change its $\theta$ ), or some other implicit assumption of Hypothesis 1 has fallen through. So, one prediction of our Hypothesis 1 is that the measurement for $b_3$ remains the same whether or not we make a measurement at $a_3$ . If we can show that this does not hold, then Hypothesis 1 does not hold.

The exact situation when we expect Hypothesis 1 to fail, due to the predictions of QM, are kind of strange. If we measure particle 1 at B, then its partner particle 2 at A, and the re-measure particle 1 at B, QM does not expect the measurement at B to change. We only expect the measurement at B to be "altered" if we look at A first . This makes observing the supposed "alteration" difficult!

However, Bell proposed the following experiment by which we can test Hypothesis 1 (and a whole class of related hypotheses). If we generate a boatload of particle pairs according to a common general scheme, and then re-calibrate A and B to various convenient values, we can predict the probability of various observations at B both with and without having "looked" at the particles at A.

Here's the setup: Generate a very large quantity of particle pairs with the first particle having uniformly-distributed $\theta_1$ , and the second having an opposite orientation $\theta_2 = \theta_1 + \pi$ . We can test uniformity by simply calibrating our measuring apparatus at random locations and making sure we get an approximately equal number of "ups" and "downs". The only way for this to happen is if the $\theta_1$ are uniform. We can test that the two particles are always opposite by checking that, when A and B are calibrated at $\phi_\text{A} - \phi_\text{B} = \pi$ apart from one another, we always measure identical readings for each particle in a pair. Set $\phi_\text{A} = 0, \phi_\text{B} = \pi$ . Alter $\phi_\text{A}$ (and only $\phi_\text{A}$ ) by a little bit. Generate another bunch of particle pairs. Now, some quantity of the particle pairs will not produce identical measurements (like our $a_3, b_3$ above). Write down this quantity $x$ . Just to double-check, reset $\phi_A$ , and alter $\phi_B$ by that same angle. Generate a bunch more particle pairs using the same mechanism. You should see that the number of unequal measurements is approximately $x$ , because the situations are symmetric (but not exactly equal, because our $\theta$ are random). Just to quadruple-check, do this a whole bunch of times to convince yourself that the number of unequal measurements is pretty much always around $x$ .

Here's the expectation: Now, change $\phi_\text{A}$ and $\phi_\text{B}$ by that small angle. In order for the problem to appear, we need to consider what might have been had we not altered $\phi_\text{A}$ or $\phi_\text{B}$ or both. If we hadn't altered either, because the measurements are all governed, under-the-covers, by $\theta$ , we would have measured identical values for all pairs. If we had only altered one or the other, we would have measured different values for $x$ pairs. If we alter both, even if none of the pairs that "change" overlap, we measure different values on $2x$ pairs. Namely, all of the pairs whose measurements "changed" at B plus all of the pairs whose measurements "changed" at A. For the remaining pairs, since their measurement didn't change at A and didn't change at B, they still give identical measurements. If there is any overlap in which pairs flipped measurements at A and B, then the number of pairs giving different measurements will be strictly less than $2x$ . To reiterate, this expectation only holds if the measurements at A and B do not effect each other. It also only holds if it is meaningful to speak of "what might have been". If the simple act of observing the spin of particle 1 at A changes the value of $\theta_2$ of its partner at B, then the situation where we make measurements at A and B need not have this particular relationship to the situation where we make only a measurement at A. For example, the act of measurement at A could change all of the $\theta$ s at B to be totally random. Or it could change the $\theta$ s at B to be the number predicted by QM. The only important thing here is that if A and B "talk", then the number of "different" measurements might be $>2x$ .

At this point, it is worth noting that the exact mechanism we proposed above is irrelevant to the argument as a whole. You can replace all of the talk of " $\theta$ " and the mechanism we proposed by which it is measured by talk of some "arbitrary locally-real variable encoding the spin info" and the inequality still holds.

1. What are the predictions made by quantum mechanics? & 3. How do they differ?

Basically, QM predicts that for certain calibrations of the equipment at A and B, we will reliably observe $>2x$ pairs that now give different measurements when we alter both $\phi_\text{A}$ and $\phi_\text{B}$ . How much different depends on complex maths that are above my pay grade. If anyone in the community has a link to a location with an explanation of this math, please comment and I will edit it in.

However, as I said above, it's largely irrelevant to Bell's result what those predictions are. Simply performing the experiment and noting that the number of pairs with different measurements is $>2x$ is enough to reject local realism, even without anything to replace it with.

Somewhat orthogonal to the predictions made by QM are the available interpretations of this result now that local realism has been tossed. This answer to a related question provides a discussion of how these interpretations tie into the results from Bell's inequality.

(1) Quantum mechanics predicts 25% or more will correlate. (2) Bell says hidden variables should correlate 33% or more of the time. (3) The difference between the two is Bell's inequality. (4) They are different things unless I misunderstand your question. Two objects are entangled if you can measure or observe one of them and instantly know something about the other one. Conservation could mean some physical quantity in an isolated system is constant. (5) Your description of local realism seems complicated. My understanding is that a particle cannot get its instructions from a distant source that would take faster than light communication. Instead the particle most likely carried the instruction from its beginning.

• $\begingroup$ What is the differences between "particle most likely carried the instruction from its beginning." and "particle most likely had its state from its beginning." $\endgroup$ –  Alberto Commented Aug 21, 2016 at 17:24
• $\begingroup$ @Alberto I said a "particle most likely carried the instructions from the beginning" but I did not say "most likely had it's state from its beginning". That would be close to the same thing. I said a "particle cannot get its instructions from a distant source" because that would take faster than light communication. That was my point. $\endgroup$ –  Bill Alsept Commented Aug 21, 2016 at 18:07

I will use polarised photons as they are simplest to understand. Also to avoid confusion, we will assume both entangled photons have the same polarisation.

What are the predictions made by quantum mechanics? What are the predictions made by local realism?

Below is a plot of the QM prediction (blue) and the local realistic model (red) correlations (y axis) versus the difference in angle between the two polarising analysers (x axis):

The quantum prediction for the correlation is equivalent to a single particle having passed through both the polarisers of the observers in the positions they are at, at the exact time the particles are received and is numerically equivalent to the equation for Malus Law ( $\cos^2(\Delta \theta)$ ) which predicts the amount of light that is transmitted trough two successive polarisers with an angle of $\Delta \theta$ between them. Since Malus Law can be represented as ( $\cos^2(|A-B|)$ ) where A in the the Angle of Alice's polariser and B is the angle of Bob's polariser, the local realistic model has the impossible task of predicting the outcome of that equation without knowing both A and B.

The implication is that each particle somehow 'knows' the position of the other polariser at the time the other particle passed though it, even though they may be separated in such a way that there is not enough time for any signal travelling at the speed of light to communicate that position. This implies non local interaction.

Bell's Theorem basically shows that any theory that relies on pre-coded information (however well hidden) and/or information passing that is limited to the speed of light, can not match the predictions of Quantum Mechanics and this is confirmed by the results of real experiments.

• $\begingroup$ It seems weird to me that Malus' Law, which was known well for 120+ years before the EPR paper, is the solution. This is the graph I've seen given as the proof, but I had never heard where the QM values were derived from. I'm guessing that QM also derives this relationship mathematically? $\endgroup$ –  jeffpkamp Commented May 9 at 18:25

Not the answer you're looking for? Browse other questions tagged quantum-mechanics quantum-entanglement locality bells-inequality epr-experiment or ask your own question .

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A much simpler introduction to the theorem, with some loss of completeness, has been prepared. You may access an html or pdf version with the links to the right.

But if we try to measure the spin at both 0 degrees and 45 degrees we have a problem. The figure to the right shows a measurement first at 0 degrees and then at 45 degrees. Of the electrons that emerge from the first filter, 85% will pass the second filter, not 50%. Thus for electrons that are measured to be spin-up for 0 degrees, 15% are spin-down for 45 degrees. Say in the illustrated experiment the left hand filter is oriented at 45 degrees and the right hand one is at zero degrees. If the left hand electron passes through its filter then it is spin-up for an orientation of 45 degrees. Therefore we are guaranteed that if we had measured its companion electron it would have been spin-down for an orientation of 45 degrees. We are simultaneously measuring the right-hand electron to determine if it is spin-up for zero degrees. And since no information can travel faster than the speed of light, the left hand measurement cannot disturb the right hand measurement. (Clarendon, Oxford, 1975), pg. 564-565. The colors were used by Wheeler in a colloquium in the Dept. of Physics, Univ. of Toronto some years ago. CELLULAR AUTOMATA A cellular automaton provides another approach to the study of the emergence of structures based on rules. One of the best known automata is the Game of Life , devised by John Conway in 1970. This example is played on a large checkerboard-like grid. One starts with a configuration of cells on the board that are populated, and then calculates the population in succeeding generations using three simple rules: Birth : an unoccupied cell with exactly 3 occupied neighbors will be populated in the next generation. Survival : an occupied cell with 2 or 3 occupied neighbors will be populated in the next generation. Death : in all other cases a cell is unoccupied in the next generation. Despite the simplicity of the rules, truly amazing patterns of movement, self-organising complexity, and more arise in this game. To the right is a Flash animation of the simplest possible configuration that changes from generation to generation but never grows or dies out. button to step from generation to generation. In this mode the number of occupied neighbors of each cell is shown. to resume playing the animation. There are many resources available on the web to explore this fascinating "game" in more detail. version of this document, Flash animations are not available from pdf. It has been proposed that these sorts of automata may form a useful model for how the universe really works. Contributors to this idea include Konrad Zuse in 1967, Edward Fredkin in the early 1980's, and more recently Stephen Wolfram in 2002. Wolfram's work in particular is the outcome of nearly a decade of work, which is described in a mammoth 1200 page self-published book modestly titled A New Kind of Science . There are two key features of cellular automata that are relevant for this discussion: The rules are always strictly deterministic. The evolution of a cell depends only on its nearest neighbors. This seems to put a cellular automaton model of Physics in conflict with Bell's Theorem, which asserts that a logical local deterministic model of the universe can not be correct. Advocates of the cellular automaton model attempt to argue that there is no essential conflict, just an apparent one. Arguments include: That the apparent randomness of quantum phenomena is only pseudo-random . To me, they seem to be re-introducing the idea of hidden variables via the back door. Plamen Petrov in one of the proponents of this argument. That there is some sort of higher-dimensional thread outside of the normal four dimensions of space and time. This "thread" will somehow allow for super-luminal connections. Wolfram and others have proposed this idea. Other Wolfram supporters have argued that the speed of light is or can be much greater than the "usual" value that we are used to. Whether or not it needs to be infinite is not clear. In the previous Bohm's Ontology of Quantum Mechanics sub-section, we saw that Bohm's attempt to keep causality ended up with a totally non-local mechanism encapsulated in a Quantum Potential . Even there, we saw at the end that there are serious problems with the model. It may be that there are even more serious problems with the Cellular Automaton model for the way the universe works. The controversy continues to be very active as of this writing (Spring, 2003). A semi-random list of further readings is: http://arxiv.org/PS_cache/quant-ph/pdf/0206/0206089.pdf http://www.math.usf.edu/~eclark/ANKOS_reviews.html http://digitalphysics.org/Publications/Petrov/Pet02m/Pet02m.htm . --> Reset password New user? Sign up Existing user? Log in Bell's Theorem Already have an account? Log in here. Bell's theorem is an important philosophical and mathematical statement in the theory of quantum mechanics . It showed that a category of physical theories called local hidden variables theory could not account for the degree of correlations between the spins of entangled electrons predicted by quantum theory. The commonly accepted conclusion of the theorem is that quantum theory is inherently nonlocal in some way, although this is a topic of intense philosophical debate. Mathematically, Bell's theorem is justified by an important lemma, the CHSH inequality . This inequality bounds the amount of correlation possible between electron spins in a hidden variables theory. The violation of the CHSH inequality in both the theory and experimental results of quantum mechanics proves the theorem. Bell's Theorem and Loopholes Chsh inequality and spin correlations. The historical importance of Bell's theorem is that it proved Einstein, Podolsky, and Rosen [EPR] incorrect in their discussion of the EPR paradox . EPR advocated a philosophy of science called local realism . This philosophy specified that interactions should not be able to communicate instantly across large distances (nonlocally) in violation of relativity, and that systems ought to have a "realistic" definite value of quantities before these quantities are measured. However, experiments measuring the correlations between spins of entangled electrons seemed to communicate spin states instantaneously between locations (nonlocally). EPR thus concluded that quantum mechanics is incomplete; there must have been some extra hidden variable that set the responses of the entangled electrons to spin measurements at the beginning of the experiment, when both electrons were generated at the same site. Bell's theorem showed that this interpretation is not true: in order to reconcile theory with experimental results of quantum mechanics, one of locality and realism must be rejected. In fact, in the popular Copenhagen interpretation of quantum mechanics, not only does quantum mechanics "communicate" spin states instantaneously, but before quantities are measured, systems do not take definite values of these quantities: both locality and realism are rejected. Bell's theorem : No theory of local realism such as a local hidden variables theory can account for the correlations between entangled electrons predicted by quantum mechanics. The experimental results of quantum mechanics have several loopholes that the past fifty years of research in quantum theory has worked to close. Below two of the major loopholes are discussed: The detection loophole It is possible that the detectors at spin measurement sites have less than 100% efficiency, and only detect highly correlated spins, allowing uncorrelated spins to remain undetected. Thus, experiments would report a higher correlation than actually exists, meaning the actual amount of correlation might be explainable by a local hidden variables theory. In the best case scenario, any detectors below 67% efficiency would not close this loophole; in the standard case, about 83% efficiency is required. Traditional Bell test experiments have ignored this loophole by postulating the fair sampling assumption , which states that the spins measured at each detector are a fair representation of the actual distribution of entangled quantum states produced. While this seems intuitively physically reasonable, it is impossible to prove. The communication loophole Since it takes a small but nonzero amount of time to actually perform spin measurements and report the result, it is possible that after one spin is measured, the detector somehow communicates the result at lightspeed to the other detector, which is able to influence the spin of the other particle at measurement time. The only way to close the communication loophole is to separate the two detectors by a large distance and perform spin measurements in such a small amount of time that light could not have traveled between the detectors. This causally separates each detector from the other's influence. In late 2015, [1] an experiment whose results were published in Nature claimed to have performed a fully loophole-free Bell experiment demonstrating violation of the CHSH inequality. Similar recent papers claim to have performed loophole-free Bell experiments for entangled photon polarizations, analogous to electron spins. It has been suggested that the late 2015 result has not quite closed all loopholes due to the possibility of communication between detectors in the past before the entangled electrons were emitted, which might have been able to correlate detectors in some way. This is known as the setting independence loophole . There exists a currently proposed experiment [2] designed to circumvent this loophole by configuring detector settings using light from two very distant galaxies so far separated that light has not traveled between the two since the big bang. As a result, the detector settings will originate from sources not in causal contact, which means that it will be impossible for the detectors to have become correlated at any point in time. The proof of Bell's theorem considers an arbitrary local hidden variables theory and shows that any such classical theory attempting to mimic the results of quantum mechanics gives measurement results constrained by an inequality called the CHSH inequality for Clauser, Horne, Shimony, and Holt, although this inequality is also often referred to slightly incorrectly as Bell's inequality . The CHSH inequality is as follows: \[|E(a,b) - E(a,b')+E(a',b)+E(a',b')| \leq 2\] where \(a\) and \(a'\) are two possible orientations for one Stern-Gerlach detector in a Bell experiment and \(b\) and \(b'\) are two possible orientations for the second detector. The value \(E(a,b)\) gives the correlation of spins along these orientations, and is defined by the expectation of the product of spin states along each direction in quantum mechanics. The formal derivation of these inequality is fairly extensive, since it introduces a possible hidden variable and then defines the expectations in terms of integrals involving this variable. However, the intuition behind it is simple: each of the expectations along orientation \(a\), \(a'\), \(b\), and \(b'\) is at most 1, because each correlation is at best 1 in a classical theory. So one has three plus signs and minus sign in a sum of terms that are bounded by one in absolute value; thus, the sum is bounded by \(2\). The tricky part of the derivation lies in manipulating the integral expressions that define the correlations to obtain nice inequalities that don't depend on the hidden variable or detector angle. An easy example of the violation of the CHSH inequality occurs in experiments measuring the spin in the entangled singlet state of two spin-\(\frac12\) particles: \[|\Psi\rangle = \frac{1}{\sqrt{2}} (|\uparrow\rangle\otimes|\downarrow\rangle - |\downarrow\rangle \otimes |\uparrow\rangle).\] The desired correlations in quantum mechanics can be found by taking the expectations of spin measurements made at two Stern-Gerlach apparatuses. Letting \(A(a)\) be a measurement performed at apparatus A in orientation \(a\) and similar, consider the following four possible measurements: \[ \begin{align} A(a) &= \hat{S}_z \otimes I\\ A(a') &=\hat{S}_x \otimes I\\ B(b) &= -\frac{1}{\sqrt{2}} I \otimes (\hat{S}_z + \hat{S}_x)\\ B(b') &= \frac{1}{\sqrt{2}} I \otimes (\hat{S}_z - \hat{S}_x) \end{align} \] where the \(b\) and \(b'\) orientations are rotated by \(135\) degrees with respect to \(a\) and \(a'\), so that with expectation values taken in the singlet state: \[ \begin{align} E(a,b) &= \langle A(a)B(b)\rangle = \frac{1}{\sqrt{2}} \\ E(a,b') &= \langle A(a)B(b')\rangle = -\frac{1}{\sqrt{2}} \\ E(a',b) &= \langle A(a')B(b)\rangle = \frac{1}{\sqrt{2}}\\ E(a',b') &= \langle A(a')B(b')\rangle = \frac{1}{\sqrt{2}} \end{align} \] The computations of the values given above are tedious but routine exercises in the formalism of spin measurement; see the quantum entanglement wiki for details of how they are performed. Substituting into the left-hand side of the CHSH inequality, one finds: \[|E(a,b) - E(a,b')+E(a',b)+E(a',b')| = \frac{4}{\sqrt{2}} = 2\sqrt{2}.\] This violates the bound of 2 predicted in a local hidden variables theory, as a result of quantum entanglement! Below, the correlation \(E(a,b)\) as a function of angle between orientations \(a\) and \(b\) is plotted. The disparity between the classical and quantum predictions is evident, especially at the \(135\) degree difference used in the above calculation, where each correlation took value \(\frac{1}{\sqrt{2}}\) where it would have correlation at most \(\frac12\) in local hidden variables theory. Optimal predictions of quantum mechanics (blue) and local hidden variables theory (red) for the correlations between electron spins in a particular entangled state. The local hidden variables theory cannot account for the degree of correlation observed [3] . Hensen, B. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature , 526, 682–686 (29 October 2015) . Gallicchio, J. Testing Bell’s Inequality with Cosmic Photons: Closing the Setting-Independence Loophole. Physical Review Letters , 112, 110405 – Published 18 March 2014 . Gill, R. Bell . Retrieved 22 December 2013, from https://en.wikipedia.org/w/index.php?curid=41434416 Problem Loading... Note Loading... Set Loading... Help | Advanced Search Quantum Physics Title: experimental tests of bell's inequalities: a first-hand account by alain aspect. Abstract: On 04 October 2022, the Royal Swedish Academy of Sciences announced that the Nobel Prize for Physics of 2022 was awarded jointly to Alain Aspect, John Clauser, and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science". What follows is an interview of Alain Aspect, conducted by Bill Phillips and Jean Dalibard, during the summer of 2022, and completed not long before the announcement of the Nobel Prize. The subject matter is essentially that for which the Nobel Prize was awarded. Comments: Accepted for publication in the topical issue "Quantum Optics of Light and Matter" of EPJD, Edts. D. Clément, P. Grangier and J. Thywissen Subjects: Quantum Physics (quant-ph) Cite as: [quant-ph]   (or [quant-ph] for this version)   Focus to learn more arXiv-issued DOI via DataCite : Focus to learn more DOI(s) linking to related resources Submission history Access paper:. Other Formats References & Citations INSPIRE HEP Google Scholar Semantic Scholar BibTeX formatted citation Bibliographic and Citation Tools Code, data and media associated with this article, recommenders and search tools. Institution arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Bahasa Indonesia Slovenščina Science & Tech Russian Kitchen 10 most interesting places in Tver Region (PHOTOS) 1. Catherine the Great's ‘Travel Palace’ in Tver  The city was founded on the Volga River in 1135. It’s 12 years older than Moscow and even competed to become the capital of Ancient Russia. Today, it is a major regional center with a population of around 414,000 people. Travelers from one capital to the other would often stop in Tver. Empress Catherine II even had a travel palace built in Tver so as to have somewhere to rest along the way. Now, it houses the Tver Regional Art Gallery. It includes artwork collections owned by Tver governors from country estates in the Tver Governorate that were nationalized after the Bolshevik Revolution. They contain works by Alexey Venetsianov, Konstantin Korovin, Arkady Plastov, Valentin Serov, Mikhail Vrubel and other famous artists. 2. Rzhev Memorial The town of Rzhev is located 120 km from Tver. From October 1941 to March 1943, some of the bloodiest battles of World War II, including the ‘Battle of Rzhev’, took place there (you can read more about the battle here ). Soviet troops lost more than 1.3 million men, including wounded, missing in action and taken prisoner. A memorial to the soldiers who fell in the battle was inaugurated in Rzhev in June 2020. At the center of the composition is a 25-meter bronze statue of a Soviet soldier whose trench coat "morphs" into a flock of cranes. The reference is to one of the most popular and poignant songs about the war titled: 'Zhuravli' ('Cranes'). It was composed by Yan Frenkel to lyrics by Rasul Gamzatov. 3. Lake Seliger Lovers of outdoor recreational activities should visit the shores of Seliger at least once in their life! This huge (260 sq. km) lake of glacial origin is home to about 30 species of fish. Hence, fishermen go there at all times of year and fish from boats, from the shore and, in winter, through ice-holes. The winding shoreline of the lake has a multitude of different hotels and campsites (as well as glamping sites), so anyone can stay there according to their preferences. And you can jump straight into the lake from the banya (bathhouse)! One of Seliger's landmarks is the charming town of Ostashkov, the largest on its shores.. Its key attractions include a Soviet local history museum, which is housed in a former church. You can also take a ride on a retro train along the Seliger - Ostashkov - Bologoye route. Seliger train 4. Nilov Monastery (St. Nilus Stolobensky Monastery) One of the main attractions in Tver Region is the Nilov Monastery, founded in the 16th century. This functioning monastery is also situated on the picturesque shores of Lake Seliger. In Soviet times, it housed a colony for young offenders, a prisoner-of-war camp, a hospital and a tourist hostel… According to legend, a hermit monk named Nil, famous for his diligent prayer, settled on the island of Stolobny on the lake. It was said that no calamities or robbers could force him off the island. After he died, other monks began to go to where his cell had stood and, eventually, they founded a monastery there. Before the Bolshevik Revolution, it was one of the most revered monasteries: Thousands of pilgrims used to visit it to worship the relics of the Venerable Nil. As part of the project ‘Russia: 85 Adventures’, we filmed a video at the monastery – you can watch it here . If, in Torzhok, you've time to eat, Pozharsky's is the place to know. Their cutlets, fried, are such a treat, Then after lunch you'll lightly go! So wrote Alexander Pushkin, who frequently traveled from St. Petersburg to Moscow to see his friend Sergei Sobolevsky. Thanks to Russia's most outstanding poet, ‘Pozharsky cutlets’ – patties of ground chicken coated in white bread croutons – became the town's most famous speciality. And they remain its calling card to this day. But, the town is famous for more than just gastronomy. The once major trading center has, today, evolved into a charming provincial town. Things to see include the picturesque scenery along the banks of the River Tvertsa, the Saints Boris and Gleb Monastery, which is virtually the oldest monastery of Ancient Russia (believed to have been founded in 1038), and the unique, wooden 17th-century Old Church of the Ascension. 6. The flooded bell tower of Kalyazin One of the region's most famous sights is the flooded bell tower of Kalyazin. The 74-meter tower protrudes out of the water not far from the shores of a reservoir. Tourists who take pictures of it are sometimes oblivious of the fact that the ruins of a once-flourishing monastery lie hidden under the water. Most of the Makaryev Monastery of the Holy Trinity was demolished in 1940, ending up in the flood zone of the Uglich hydroelectric power station on the Volga River and the Uglich Reservoir. What remains now as a reminder of the monastery are the bell tower, which was recently restored and re-whitewashed and also a set of frescoes miraculously rescued from the monastery. You can read more about them here . 7. The source of the Volga It's hard to believe that this spring and stream in the Valdai Heights are the place where one of the world's biggest rivers (and the biggest in Europe) rises. Next to the spring stand a chapel and a footbridge with a plaque – an ideal spot for a souvenir selfie! In Ancient Rus’, the River Volga was always held in special esteem – it was described as “Mother Volga”, a multitude of towns were built along it and it provided food for a large number of Russian regions and continues to do so to this day. This is why pilgrimages have been made to its source for several centuries now. Back in the 17th century, a monastery stood there, but it burnt down and was never restored. A new one was, however, built in 1912 – the Olginsky Convent. 8. Shirkov Pogost This spot on the shores of Lake Vselug (today part of the Volga River) is dubbed the "Kizhi of the Tver Region". Like the famous Kizhi on Lake Onega , Shirkov Pogost is of interest because of its multi-tiered wooden church – in this case, the church of John the Baptist, a masterpiece of Russian wooden architecture, which was built in 1697. According to one legend, the Pogost was named in honor of the Shirkov brothers, merchants who had the church built. They were taking two icons of John the Baptist from Novgorod to Moscow. On this spot, they laid down the sacred images and decided to rest, but they could not pick them up again – and, so, decided to build a church there without using a single nail! 9. Vyshny Volochyok The town got its name from the word ‘volok’ (‘portage’) – in other words, various watercraft were carried across a section of dry land between two bodies of water there. True enough, Vyshny Volochyok was on the watershed of the basins of the Baltic (Tsna River) and the Caspian (Tvertsa River, a tributary of the Volga). Under Peter I, the first artificial canal in Russia was built between the two rivers. Vyshny Volochyok was always an important staging post on the route from Moscow (and Central Russia) to St. Petersburg. Because of its convenient location, there were many factories and production plants there, from garment factories to glass and woodwork facilities. Volochyok is also famous for the manufacture of ‘valenki’ (‘felt boots’) and it even has a museum devoted to this footwear and to the art of wool hand felting. A host of old factory buildings, as well as houses that once belonged to affluent merchants in a great variety of styles, have survived in the town. A roadside imperial palace was also built there, which, today, houses a school. 10. Konakovo Until 1929, the village was called Kuznetsovo, after which it was renamed in honor of Porfiry Konakov, a participant in the 1905 Russian Revolution. The area is famous for its china. The Konakovo Pottery Factory was founded as early as 1809 and is one of the oldest in Russia. It is no longer in operation, but local craftspeople are using the legacy of its artistic workshop to revive production today. Items can be bought as a souvenir in Tver Region. But, today's tourists know the location more as a fashionable riviera. Konakovo and nearby Zavidovo are popular places for recreational activities and water sports. People go sailing, wakesurfing and wakeboarding and there are a multitude of hotels and glamping sites for holiday stays. And the close proximity to the M11 motorway makes the location even more attractive (particularly to Muscovites, who can get there in just over an hour!).  Dear readers, Our website and social media accounts are under threat of being restricted or banned, due to the current circumstances. So, to keep up with our latest content, simply do the following: Subscribe to our Telegram channel Subscribe to our weekly email newsletter Enable push notifications on our website Install a VPN service on your computer and/or phone to have access to our website, even if it is blocked in your country If using any of Russia Beyond's content, partly or in full, always provide an active hyperlink to the original material. to our newsletter! Get the week's best stories straight to your inbox Pozharsky cutlets: Cooking the most tender and juicy croquettes EVER! (RECIPE) Toll roads in Russia: How they work and how to use them How to travel to Russia with a camper: Tips for novices This website uses cookies. Click here to find out more. Tver Oblast Overview Map Directions Satellite Photo Map Overview Map Directions Satellite Photo Map Tap on the map to travel Popular Destinations Type: State with 1,280,000 residents Description: administrative division (oblast) in central Russia Neighbors: Moscow Oblast , Novgorod Oblast , Pskov Oblast , Smolensk Oblast , Vologda Oblast and Yaroslavl Oblast Categories: oblast of Russia and locality Location: Central Russia , Russia , Eastern Europe , Europe View on Open­Street­Map Tver Oblast Satellite Map Popular Destinations in Central Russia Explore these curated destinations. 401 IMAGES Bell's inequality applied to experiments in which Alice and Bob each (a) Bell inequality experiment. A source emits a composite system and Imaging setup to perform a Bell inequality test in images. A BBO First experiment to explore the Bell Inequality The meaning of Bell's inequality Quantum Entanglement Bell Tests Part 1: Bell's Inequality (My Best Explanation) VIDEO The EPR Paradox & Bell's inequality explained simply Bell's Inequality Bell's Inequality II 2017 Andrew Carnegie Lecture: Professor Alain Aspect Discussing The Bell Curve with Claude S. Fischer (Professor of Sociology Bell's Inequality experiment plus interviews COMMENTS Bell test A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism.Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables (called "hidden ... Bell's theorem Bell then showed that quantum physics predicts correlations that violate this inequality. Multiple variations on Bell's theorem were put forward in the following years, using different assumptions and obtaining different Bell (or "Bell-type") inequalities. The first rudimentary experiment designed to test Bell's theorem was performed in 1972 by ... How Bell's Theorem Proved 'Spooky Action at a Distance' Is Real Thus the labs will obtain opposite results when measuring along different axes at least 33% of the time; equivalently, they will obtain the same result at most 67% of the time. This result — an upper bound on the correlations allowed by local hidden variable theories — is the inequality at the heart of Bell's theorem. Above the Bound Bell's Theorem This was a difficult experiment, requiring 200 hours of running time, much longer than in most later tests of Bell's Inequality, which were able to use lasers for exciting the sources of photon pairs. Since then, several dozen experiments have been performed to test Bell's Inequalities. PDF 22.51 Lecture slides: EPR paradox, Bell inequalities In a complete theory there is an element corresponding to each element of reality.A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non- commuting operators, the knowledge of one ... PDF 51 Hidden variables and Bell's inequalities Bell's inequalities and experiment If we find an experiment that violates this inequality then we have to throw out deterministic local hidden variable theories e.g., the idea that the photon has a variable that it carries with it that determines the result of the polarization measurement Experiments do violate this inequality Lecture Notes: Bell's inequality and quantum entanglement Lecture Notes: Bell's inequality and quantum entanglement Download File DOWNLOAD. Course Info Instructor Prof. David Kaiser; Departments Science, Technology, and Society; Physics; As Taught In Fall 2020 Level Undergraduate. Topics Humanities. History. History of Science and Technology ... How the Bell tests changed quantum physics This fascination led Aspect to envisage experimental tests of Bell's inequalities closer to Bell's ideal scheme than the first test 2 realized by Stuart J. Freedman and John F. Clauser in 1972 ... Bell's Inequalities Experiments Abstract. The tests of Bell's inequalities were the first and most important experiments in quantum foundations. This chapter presents their origins, historical background, conceptual formulation, and requirements. In the 1960s, thanks to a paper by John Stuart Bell, it became possible to turn thought experiments conceived in the early prewar ... The EPR Paradox and Bell's Inequality Does Bell's Inequality rule out local theories of quantum mechanics? In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. This so-called "EPR Paradox" has led to much subsequent, and still ongoing, research. ... Bell's theorem 1 Historical background; 2 The EPR argument for pre-existing values; 3 Bell's inequality theorem; 4 Bell's theorem; 5 The CHSH-Bell inequality: Bell's theorem without perfect correlations; 6 Bell's definition of locality; 7 Experiments; 8 Bell's theorem and non-contextual hidden variables; 9 Bell's theorem without inequalities; 10 Controversy and common misunderstandings. 10.1 Missing the ... Bell's Theorem (33) but in good agreement with the quantum mechanical prediction Eq. (34). This was a difficult experiment, requiring 200 hours of running time, much longer than in most later tests of Bell's Inequality, which were able to use lasers for exciting the sources of photon pairs. Several dozen experiments have been performed to test Bell's ... quantum mechanics If it is higher than the Bell inequality then that contradicts LR. There is not observable effect of one particle on another when measuring an entangled particle. It is just in the statistics. The real meat is in Bell's proof that LR implies Bell's inequality. This means that experiments that contradict Bell's inequality also contradict LR. Bell's Theorem PROVING BELL'S INEQUALITY. We shall be slightly mathematical. The details of the math are not important, but there are a couple of pieces of the proof that will be important. ... The result of the experiment is that the inequality is violated. The first published experiment was by Clauser, Horne, Shimony and Holt in 1969 using photon pairs. The ... Bell's Theorem In late 2015, an experiment whose results were published in Nature claimed to have performed a fully loophole-free Bell experiment demonstrating violation of the CHSH inequality. Similar recent papers claim to have performed loophole-free Bell experiments for entangled photon polarizations, analogous to electron spins. Experimental tests of Bell's inequalities: A first-hand account by View a PDF of the paper titled Experimental tests of Bell's inequalities: A first-hand account by Alain Aspect, by William D. Phillips and Jean Dalibard. On 04 October 2022, the Royal Swedish Academy of Sciences announced that the Nobel Prize for Physics of 2022 was awarded jointly to Alain Aspect, John Clauser, and Anton Zeilinger "for ... Administrative divisions of Tver Oblast 1,411. Administratively, Tver Oblast is divided into two urban-type settlements under the federal government management, five cities and towns of oblast significance, and thirty-six districts . In terms of the population, the biggest administrative district is Konakovsky District (87,125 in 2010), the smallest one is Molokovsky District (5,235 ... 10 most interesting places in Tver Region (PHOTOS) 1. Catherine the Great's 'Travel Palace' in Tver. The city was founded on the Volga River in 1135. It's 12 years older than Moscow and even competed to become the capital of Ancient Russia ... Tver Oblast Map Tver Oblast is a region in Central Russia, which borders Smolensk Oblast to the southwest, Pskov Oblast to the west, Novgorod Oblast to the north, Vologda Oblast to the northeast, Yaroslavl Oblast to the east, and Moscow Oblast to the southeast. Photo: Belliy, CC BY-SA 4.0. Photo: Florstein, CC BY-SA 3.0. Randonaut Trip Report from Tver, Tver' Oblast (Russian Federation) Posted by u/therealfatumbot - 1 vote and no comments essay homework paper phd project resume review study thesis