User:Caroline Thompson/Bell's theorem

Bell's theorem was first proposed in 1964 by the physicist John S. Bell, and provides the basis for experimental tests of local realism versus quantum mechanics. It states that a certain inequality has to be satisfied if the world works according to the principle of "local realism" but can be violated by quantum mechanics. The history and implications of the theorem are discussed in the EPR paradox and interpretation of quantum mechanics pages, and in many popular books, some of which are listed at the end of this article.

Introduction
Although the theorem is named after John Bell, he himself never formulated it as such and, since publication of his paper in 1964, a number of different inequalities have been derived, all termed "Bell inequalities". They all purport to make the same assumptions about local realism -- that even quantum-level objects have well defined parameters and that distant objects do not exchange information instantaneously -- though some in fact make additional assumptions and so are less general than others. The inequalities concern measurements made on pairs of particles that have interacted and then separated. They place limits on the possible values of the "quantum correlation" that can be obtained, on the assumption that the only possible factors causing the correlation are shared "hidden variables", set at the souce -- the point of interaction. In quantum mechanics, the existence of these hidden variables associated with individual particle pairs is denied, the correlation being instead due to quantum entanglement of the whole ensemble. The correlation is predicted to exceed Bell's limits.

Bell test experiments to date suggest that Bell's inequality is violated, and the general belief in the physics community is that they prove quantum mechanics is correct and local realism does not hold. Nevertheless, the issue is far from being conclusively settled. The experiments themselves can be challenged on the grounds that they make various assumptions which create loopholes that open the way for alternative interpretations. Designing experiments which more conclusively determine whether or not the Bell inequality holds remains a very active area of research (Shimony, 2004), as does the search for local hidden variable theories that can explain existing Bell test violations.

Bell test experiments


If the world has underlying local hidden variables, then Bell's inequalities must be obeyed by the "coincidence counts" from a Bell test experiment such as the optical one shown in the diagram. Pairs of particles are emitted as a result of a quantum process, analysed with respect to some key property such as polarisation direction, then detected. The setting (orientations) of the analysers are selected by the experimenter. Several different subexperiments are conducted, one set for each of up to four choices of setting. In some Bell tests the analysers only have a single output channel. For these, extra subexperiments are needed in which polarisers are absent on one or other or both sides, and a Bell inequality designed for the purpose (the CH74 inequality) is used.

Existence of hidden variables
Under local realism, it is assumed that the photons in a Bell test experiment have definite properties such as polarisation direction even before they are detected. This contrasts with quantum mechanics, which says that the particles emitted from a quantum event exist in a superposition of polarisation states until detected. Bell's original paper concerned "spin-½" particles rather than photons, and he assumed each particle had definite spin, the "hidden variable" &lambda;. He assumed that knowing &lambda; and the detector setting, the outcome was completely determined.

Stochastic hidden variables
It was later found (Bell, 1971; Clauser, 1974) that in real experiments it was best to recognise the fact that the components of &lambda; (and there can be many) fall into two sets, those that describe the quantum state of the source at the moment of emission and "others", primarily associated with the polarisers and detectors. It is the former that play the logical role of hidden variables, the latter being effectively random. They can be averaged out, leaving the non-random ones as "stochastic hidden variables", determining (together with the polariser settings) not the actual outcomes but just their probabilities. Clauser and Horne termed a theory based on this kind of hidden variable an "Objective Local Theory", or OLT. It is this kind that is used in the later (definitive) derivations of the inequalities used in actual experiments.

Locality
The other vital assumption is "locality" -- once the particles are separated they do not have any influence on each other, and the two detectors do not affect each other. Only "local" factors can affect each particle. In some experiments elaborate precautions have been taken to ensure that if there were any influence of one side of the experiment on the other then it could only be one that acted faster than the speed of light.

Separability
Given these assumptions, it can be shown that the joint probability of detection of both particles A and B in their respective '+' channels in a Bell test experiment is


 * (1) pAB(a, b, &lambda;) = pA(a, &lambda;)pB(b, &lambda;)

when the detector settings are a and b respectively and the source at the instant of emission was in the state &lambda;. Assuming that the source takes states &lambda; with probability &rho;(&lambda;), it follows that the probability of a coincidence is


 * (2) $$\mbox{P}_{AB}(a,b) = \int d\lambda \rho(\lambda) \mbox{p}_A(a,\lambda)\mbox{p}_B(b,\lambda)$$

The expression (2) is said to be "separable" -- the individual probabilities in the integrand are multiplied as in equation (1) to give the joint proability. This contrasts with the quantum mechanical "nonseparable" formula, equation (9) below.

Bell's original inequality
The inequality that Bell derived in 1964 was:


 * (3) 1 + E(b, c) &ge; |E(a, b) - E(a, c)|,

where E is the "quantum correlation" (the expectation value of the product of the outcomes) of the particle pairs and a, b and c settings of the apparatus (in practice, orientations of the analysing polariser) in a Bell test experiment. Outcomes are coded as +1 for the + channel, -1 for the - channel.

This inequality is not used in practice. For one thing, it is only true for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and -1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction.

Optical experiments, incidentally, can scarcely be said to have confirmed this last prediction, since even with analysers parallel there are a great number of non-detections due to the inefficiency of the detectors.

The CHSH inequality
In 1969 Clauser, Horne, Shimony and Holt (Clauser, 1969) derived the slightly more general "CHSH" inequality, though in the proof they gave initially it would appear to be valid only for two-outcome setups. It was only later, in 1971, that Bell published an improved derivation, valid even when there are non-detections. This derivation will be found on the "CHSH inequality" page. In the 1969 paper, incidentally, the inequality that was in fact recommended was not the CHSH inequality (which applies to two-channel experiments) but an adaptation to the single-channel case, the authors dismissing the former as impractical (Thompson, 2004). (The single-channel test suggested in 1969 was in fact the same one that is now associated with Clauser and Horne's 1974 article, referred to in the page on the subject as the CH74 test.)

The CHSH inequality is:


 * (4a)  &minus;2 &le; S &le; 2,

where
 * (4b) S =  E(a, b) &minus; E(a, b&prime;) + E(a&prime;, b) + E(a&prime; b&prime;)

and E(a, b) is, as before, the expectation value of the quantum correlation when the analyser settings are a and b:


 * (5) E(a, b) = p++(a, b) + p--(a, b) &minus; p+-(a, b) &minus; p-+(a, b).

This inequality is widely used, though it did not come into favour until after Aspect had inaugurated the idea in 1982 (Aspect, 1982). The reason for reluctance to use it at first was that, strictly speaking, each term E, being formed from sums and differences of four probabilities, ought to be estimated using unbiased estimates of those probabilities. The probabilities ought to be estimated as ratios of the observed frequencies of coincidences to the number N of emitted pairs. This number, however, is not known in real experiments and is, in any case, very large compared to the coincidence counts. If it were to be used, no optical experiment could hope to violate the inequality. So in practice the estimate


 * (6) E = (N++ + N-- &minus; N+- &minus; N-+)/(N++ + N-- + N+- + N-+)

is used. This estimate can only be justified if the sample of detected pairs is a fair sample of those emitted. Local realists consider that there is good reason to expect that the sample is not in fact fair, given the low efficiencies of the detectors and the likely characteristics of the apparatus. The search for more efficient detectors or for other ways of achieving a "loophole-free" experiment continues.

The CH74 inequality
The inequality derived by Clauser and Horne in 1974 depends only on ratios of observed counts, with no presupposition that they represent probability estimates. Knowledge of N, the number of pairs emitted, is not necessary (Clauser, 1974; Clauser, 1978), nor is it important to have high efficiency detectors. Just one assumption is needed in addition to "separability". It is assumed that there is "no enhancement" -- that when a polariser is inserted the probability of detection of a photon never increases. The derivation is given on the page "Clauser and Horne's 1974 Bell test".

The inequality is


 * (7a) S &le; 0,

where
 * (7b) S = (N(a, b) - N(a, b&prime;) + N(a&prime;, b) + N(a&prime;, b&prime;) - N(a&prime;, &infin;) - N(&infin;, b)) / N(&infin;, &infin;).

The Wigner-d'Espagnat inequality
The Wigner-d'Espagnat inequality is almost equivalent to Bell's original inequality, though it is expressed in terms of probabilities, not expectation values of products of outcomes.

Just three different settings are used for the detectors, which are taken as giving definite either + or - outcomes. The inequality is:


 * (8) P++(a, b) &le; P++(a, c) + P++(c, b),

where P is the probability of a coincidence.

It is valid only when there are either no non-detections at all (or, failing this, the detected pairs are a "fair sample" of those emitted), as can readily be seen from C. H. Thompson's Chaotic Ball model (Thompson, 1996). For this reason it remains just an instructive thought-experiment. It was used as such by Bell in his "Bertlmann's socks" paper (Bell, 1981).

Other inequalities
In 1989 Greenberger, Horne, and Zeilinger produced an alternative to the Bell setup, known as the GHZ experiment. It uses three observers and three electrons, and is designed to be able to distinguish hidden variables from quantum mechanics in a single set of observations (Greenberger, 1990, 1993).

In 1993 Hardy proposed a situation where nonlocality can be inferred (in ideal conditions) without using inequalities (Hardy, 1993).

Comparison with the quantum-mechanical prediction
To complete the proof of Bell's theorem it is necessary to demonstrate that quantum mechanics makes a prediction that violates a "Bell inequality". For a derivation of the quantum-mechanical prediction see the page "Quantum mechanical Bell test prediction", which is based on section 2 of an encyclopaedia article written by Abner Shimony (Shimony, 2004), one of the authors of the original Clauser, Horne, Shimony and Holt article (Clauser, 1969) after which the CHSH Bell test is named.

In Bell's spin-&frac12; setup we assume the two sides have opposite spins for the same hidden variable value, and spin is the ordinary geometrical concept at least insofar "up" is geometrically opposite to "down". The quantum-mechanical prediction for the probability for '++' coincidences is in this case:


 * (9) P++(a, b) = &frac12; sin2&phi;/2,

where &phi; = (b &minus; a), with similar predictions for the other coincidence categories.

In an optical experiment we deal instead with "vertical" and "horizontal" polarisation, i.e. orthogonal instead of opposite directions, and assume in practice that the polarisation directions on the two sides are the same, not opposite. The prediction for '++' coincidences is:


 * (10) P++(a,  b) = &frac12; cos2&phi;.

Let us consider a two-channel optical experiment. The prediction for the quantum correlation is, from (10) and the similar expressions for the other three coincidence categories,


 * (11) E(a, b) = &frac12; cos2 &phi; + &frac12; cos2 &phi; - &frac12; sin2 &phi; - &frac12; sin2 &phi;
 * = cos2 &phi; - sin2 &phi;
 * = cos 2&phi;.

Now the test statistic for the CHSH inequality is (equation (4b)):


 * S = E(a, b) &minus; E(a, b&prime;) + E(a&prime;, b) + E(a&prime;, b&prime;).

Let us choose as the detector settings a = 0, a&prime; = &pi;/4, b = &pi;/8, b&prime; = 3 &pi;/8 and evaluate the quantum mechanical prediction for S. We have


 * E(a, b) = E(a&prime;, b) = E(a&prime;, b&prime;) = cos (&plusmn;&pi;/4) = 0.707,

and
 * E(a, b&prime;) = cos 3&pi;/4 = &minus;0.707,

giving S = 2.828.

This exceeds the limit of 2 that must, by inequality (4a), be obeyed by any local hidden variable theory, and thus confirms Bell's theorem.

Experimental applications of Bell's theorem
The first application of Bell's theorem was in 1972, by Freedman and Clauser (Freedman, 1972). This was a single-channel experiment, using an atomic radiative cascade source and "pile-of-plates" polarisers. The results violated the Freedman inequality -- an inequality that can be derived from the full version of the CH74 inequality when the source is rotationally invariant (Clauser, 1978).

Other applications followed during the 1970s, then in 1981-2 Alain Aspect and his team in Orsay, Paris, conducted the three experiments for which he is now well known (Aspect, 1981-2). These experiments used a similar source, and the first and last also used similar polarisers, to Freedman and Clauser. The source was not so convincingly rotationally invariant, though, so the general form of the CH74 test was used. In the remaining experiment, with two-channel polarisers, the CHSH test was used, this being the first application of the test.

All three of Aspect's experiments violated their respective Bell inequalities and have been taken by the majority of physicists as convincing evidence of quantum entanglement and the impossibility of any local realist (local hidden variable) model for quantum events. Experimental conditions were not perfect, however. There were large numbers of "accidentals", and it is possible that the subtraction of these -- justified by Aspect using quantum-mechanical arguments -- biased the analysis (Thompson, 2003). The two-channel experiment, for which the subtraction of accidentals was not critical (Aspect, 1985), required the fair sampling assumption. Despite Aspect's best efforts to check that the sample was fair, this, as has been known since 1970 (Pearle, 1970), represents a possible loophole. The last experiment is probably the best known, having used an acousto-optical system to switch the photon direction on each side to one of two polarisers, thus blocking the locality loophole.

Theoretical work and experimental tests have since continued in two directions, the purely scientific search for the final conclusive (loophole-free) test (Fry, 1995 & 2002, Sanchez, 2004), and the development of applications of quantum entanglement. In the latter, Bell tests (generally CHSH tests) are conducted in order to check that the particles used are entangled, but assumptions such as fair sampling are made with little question.

The majority of Bell test experiments now use parametric down-conversion (PDC) sources to generate the entangled photons, often using the efficient methods devised by Paul Kwiat and his team (Kwiat, 1995, 1999).

Implications of violation of Bell's inequality
If quantum entanglement is indeed a fact of this universe, then, as is widely recognised, a major revision of our concept of reality is required. New interpretations of quantum mechanics as alternatives to the conventional Copenhagen interpretation have been proposed as possible ways of smoothing over the difficulties. These include the many worlds and consistent histories interpretations.

As far as the experimental evidence is concerned, though, the various loopholes mean that there have as yet been no conclusive violations of a Bell inequality. If local realists are correct and a local hidden variable theory exists, the experiments to date have no special implications and science can revert to the intuitive approaches prevalent a century ago.

Some advocates of the hidden variables idea prefer to accept the opinion that experiments have ruled out local hidden variables. They are ready to give up locality (and probably also causality), explaining the violation of Bell's inequality by means of a "non-local" hidden variable theory, in which the particles exchange information about their states. This is the basis of the Bohm interpretation of quantum mechanics. It is, however, considered by most to be unconvincing, requiring, for example, that all particles in the universe be able to instantaneously exchange information with all others.

Finally, one subtle assumption of the Bell inequalities is counterfactual definiteness. The derivation involves talking about two or more alternative measurements that cannot in practice all be performed on any given individual particle, since the act of taking the measurement changes the state of the particle. Under local realism the difficulty is readily overcome, so long as we can assume that the source is stable, producing the same statistical distribution of states for all the subexperiments. If this assumption is felt to be unjustifiable, though, one can argue that Bell's inequality is unproven. In the Everett many-worlds interpretation, the assumption of counterfactual definiteness is abandoned, this interpretation assuming that the universe branches into many different observers, each of whom measures a different observation.