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Bell's theorem

In foundational studies of quantum mechanics, Bell's theorem refers to a class of correlation inequalities that hold under local realism. The inequalities are named after John Bell, who in a series of papers in the mid-1960s critically examined von Neumann's proposed proof of the non-existence of hidden variables. More significantly, Bell's theorem offered a way of quantifying some of the concepts associated with the EPR paradox and eventually provided experimental tests of quantum entanglement versus local realism.

Bell considered a setup in which two observers, now commonly referred to as Alice and Bob, perform independent measurements on a system S prepared in some fixed state. Moreover, on each trial, Alice and Bob can choose between various detector settings; after repeated trials Alice and Bob collect statistics on their measurements and correlate the results. In one version of this setup, Alice can choose between two detector settings to measure one of X_A or Y_A, and Bob can choose between detector settings to measure either X_B or Y_B. Each measurement has one of two possible outcomes +1, −1.

As an example, consider a composite system consisting of two electrons prepared in a special state, one of which is sent to Alice and the other one to Bob. Alice and Bob then each measure the spin of their electron along one of two perpendicular axes.

There are two key assumptions in Bell's analysis: (1) each measurement reveals an objective physical property of the system (2) a measurement taken by one observer has no effect on the measurement taken by the other.

In the version of the inequality due to Clauser, Horne, Shimony and Holt (called the CHSH form):

$(1) \quad \mathbf{C}(X_A, X_B) + \mathbf{C}(X_A, Y_B) + \mathbf{C}(Y_A, X_B) - \mathbf{C}(Y_A, Y_B)\leq 2,$

where C denotes correlation.

Experimental tests of Bell inequalities support the failure of local realism, and in particular, that some of unexpected correlations suggested by the EPR thought experiment do in fact occur. However, by the no-communication theorem, it is impossible for Alice to communicate information to Bob (or vice versa) in violation of relativity.

Contents

1 Correlation

2 Comparison to quantum mechanical prediction

3 Bell test experiments

Correlation

In statistics, the correlation coefficient of random variables X, Y is

$\frac{1}{\sigma_X \sigma_Y} \bigg(\operatorname{E}(X Y) - \operatorname{E}(X) \operatorname{E}(Y)\bigg),$

where σ_X is the square root of the variance of X. However, in this article, we will refer to the closely related, but unnormalized quantity

$\mathbf{C}(X,Y) = \operatorname{E}(X Y)$

as the correlation. To estimate a correlation, we take observations on independent repeated trials of the pair (X, Y). In this case, the law of large numbers says that under (relatively minor) technical assumptions

$(2) \quad \mathbf{C}(X,Y) = \lim_{N \rightarrow \infty} \frac{1}{N} (X_1 Y_1 + \cdots + X_N Y_N)$

almost surely. Equation (2) makes sense for any sequence of measured values X_n, Y_n. We use (2) to define the correlation provided that the limit in (2) exists and is "robust", meaning that it exists for "enough" subsequences (and with the same value).

We can now state the following "theorem", although we have not produced here a mathematical theory in which this theorem follows deductively. See for example (Bell, 1971).

Bell's theorem. The CHSH inequality (1) holds under the local realist assumptions above.

Consider an infinite sequence of Bell test trials. This consists of:

An infinite sequence of values X_Aⁿ, Y_Aⁿ, X_Bⁿ, Y_Bⁿ, where n denotes the trial number. These values correspond to physical properties that can be revealed by the measurement.

Two infinite sequences of measurement choices, one each for Alice and Bob.

The values of the each one of the correlation expressions

$\mathbf{C}(X_A, X_B), \mathbf{C}(X_A,Y_B), \mathbf{C}(Y_A, X_B), \mathbf{C}(Y_A,Y_B)$

is estimated by extracting appropriate measurement subsequences from the entire run. However, the robustness assumption can be used to conclude that the correlation expressions are equal to those obtained by taking the limit of the averages on the entire run, including those values that were not selected for measurement.

By the locality assumption for each trial, at least one of

$X_B^n + Y_B^n, \quad X_B^n - Y_B^n$

is zero regardless of which of the variables X_A or Y_A is measured by Alice. Thus

$\sum_{n=1}^N \bigg(X_A^n \ X_B^n + X_A^n \ Y_B^n +Y_A^n \ X_B^n - Y_A^n \ Y_B^n\bigg) \leq 2 N,$

since each summand can be regrouped as

$X_A^n \ X_B^n + X_A^n \ Y_B^n +Y_A^n \ X_B^n - Y_A^n \ Y_B^n = X_A^n (X_B^n + Y_B^n)+ Y_A^n (X_B^n - Y_B^n) \leq 2.$

Thus

$\mathbf{C}(X_A, X_B)+ \mathbf{C}(X_A,Y_B)+ \mathbf{C}(Y_A, X_B) - \mathbf{C}(Y_A,Y_B) =$

$= \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n=1}^N X_A^n \ X_B^n + \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n=1}^N X_A^n \ Y_B^n + \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n=1}^N Y_A^n \ X_B^n - \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{n=1}^N Y_A^n \ Y_B^n$

$= \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N \bigg(X_A^n \ X_B^n + X_A^n \ Y_B^n +Y_A^n \ X_B^n - Y_A^n \ Y_B^n\bigg)$

$\leq \lim_{N \rightarrow \infty} \frac{1}{N} \ 2 N = 2. \quad$

Remark 1. There remain a number of issues regarding the experimental estimation of the correlations. For instance, each trial n falls into exactly one of the following subsequences:

A measures X_Aⁿ, B measures X_Bⁿ,
A measures X_Aⁿ, B measures Y_Bⁿ,
A measures Y_Aⁿ, B measures X_Bⁿ,
A measures Y_Aⁿ, B measures Y_Bⁿ.

We need to insure that each expression has the same value when taken on each one of the above subsequences.

In fact, there other possibilities that are ignored in the above list. One such possibility regards detection failures; see loopholes. Also see section 4.1 of (Redhead, 1987) for further discussion of these assumptions and caveats on the proof of Bell's theorem and in particular, what is not assumed about the correlation limits. Section 4.2 of this reference also discusses the relation between various properties such as local counterfactual definiteness, locality and determinism. See also section 4.3 of (van Frassen, 1991). A more recent treatment mostly formulated within a particular interpretative framework (consistent histories) is given in (Griffiths, 2002).

Remark 2. The validity of the correlation inequality (1) still holds if the variables X_A, Y_A, X_B, Y_B are allowed to take on any real values between -1, +1 (in addition to the local realist assumptions, of course). Indeed, the relevant idea is that each summand in the above average is bounded above by 2. This is easily seen to be true in the more general case:

$X_A \ X_B + X_A \ Y_B + Y_A \ X_B - Y_A \ Y_B =$

$= X_A (X_B + Y_B) +Y_A (X_B - Y_B) \quad$

$\leq \bigg|X_A (X_B + Y_B) +Y_A (X_B - Y_B) \bigg| \quad$

$\leq \bigg|X_A (X_B + Y_B)\bigg| +\bigg|Y_A (X_B - Y_B)\bigg|$

$\leq |X_B + Y_B| +| X_B - Y_B| \leq 2.$

To justify the upper bound 2 asserted in the last inequality, without loss of generality, we can assume that

$X_B \geq Y_B \geq 0.$

In that case

$|X_B + Y_B| +| X_B - Y_B| =X_B + Y_B + X_B - Y_B = 2 X_B \leq 2.$

Also see local hidden variables.

Comparison to quantum mechanical prediction

To apply Bell's theorem we will show that quantum mechanics makes a prediction that violates a "Bell inequality" in the setup considered in the EPR thought experiment. In order to do this, we first need to show how to compute correlations of quantum mechanical observables.

In the usual quantum mechanical formalism, observables X, Y are represented as self-adjoint operators on a Hilbert space. To compute the correlation, assume that X, Y are represented by matrices in a finite dimensional space and that X, Y commute; this special case suffices for our purposes below. We then use the von Neumann measurement postulate: a measurement of an observable X in system state φ produces a distribution of real values in which the probability of observing λ is

$\|\operatorname{E}_X(\lambda) \phi\|^2$

(where E_X(λ) is the eigenspace corresponding to λ) and the system state immediately after the measurement is

$\|\operatorname{E}_X(\lambda) \phi\|^{-1} \operatorname{E}_X(\lambda) \phi.$

From this, we can show that the correlation of X, Y in a pure state ψ is

$\langle X Y \rangle = \langle X Y \psi \mid \psi \rangle.$

We apply this fact in the context of the EPR paradox. Let us consider the spin observables for an electron along the x and z axes. The observables are represented by the 2 × 2 self-adjoint matrices:

$S_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

$S_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$

These are the Pauli spin matrices normalized so that the corresponding eigenvalues are +1, −1. As is customary, we denote the eigenvectors of S_x by

$\left|+x\right\rang, \quad \left|-x\right\rang.$

Let φ be the spin singlet state for a pair of electrons discussed in the EPR paradox. This is a specially constructed state described by the following vector in the tensor product

$\left|\phi\right\rang = \frac{1}{\sqrt{2}} \bigg(\left|+x\right\rang \otimes \left|-x\right\rang - \left|-x\right\rang \otimes \left|+x\right\rang \bigg).$

Now let us apply the CHSH formalism to the observables:

$X_A = S_z \otimes I$

$Y_A = S_x \otimes I$

$X_B = - \frac{1}{\sqrt{2}} \ I \otimes (S_z + S_x)$

$Y_B = \frac{1}{\sqrt{2}} \ I \otimes (S_z - S_x).$

The operators X_B, Y_B correspond to B's spin measurements along skew directions in the xz-plane. Note that the operators subscripted with A commute with those subscripted with B so we can apply our calculation for the correlation. In this case, we can show that the CHSH inequality fails. In fact, a straightforward calculation shows that

$\langle X_A X_B \rangle = \langle X_A Y_B \rangle =\langle Y_A X_B \rangle = \frac{1}{\sqrt{2}},$

and

$\langle Y_A Y_B \rangle = - \frac{1}{\sqrt{2}}.$

so that

$\langle X_A X_B \rangle + \langle X_A Y_B \rangle + \langle Y_A X_B \rangle - \langle Y_A Y_B \rangle = \frac{4}{\sqrt{2}} > 2.$

Thus, if the quantum mechanical formalism is correct, then the system consisting of a pair of entangled electrons cannot satisfy the principle of local realism. Note that $2 \sqrt{2}$ is indeed the upper bound for quantum mechanics, it's called Tsirelson's bound. The operators giving this maximal value are always isomorphic to the Pauli matrices.

The next sections consider experimental tests to see whether the Bell inequalities required by local realism hold up to the empirical evidence.

Bell test experiments

Main article: Bell test experiments.

Bell's inequalities are tested by "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.

Bell test experiments to date overwhelmingly suggest that Bell's inequality is violated. Indeed, a table of Bell test experiments performed prior to 1986 is given in 4.5 of (Redhead, 1987). Of the thirteen experiments listed, only two reached results contradictory to quantum mechanics; moreover, according to the same source, when the experiments were repeated, "the discrepancies with QM could not be reproduced".

Nevertheless, the issue is not conclusively settled. According to Shimony's 2004 Stanford Encyclopedia overview article

"Most of the dozens of experiments performed so far have favored Quantum Mechanics, but not decisively because of the 'detection loopholes' 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."

These loopholes are discussed in the Bell test experiments article.

References

A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981)
A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
A. Aspect and P. Grangier, About resonant scattering and other hypothetical effects in the Orsay atomic-cascade experiment tests of Bell inequalities: a discussion and some new experimental data, Lettere al Nuovo Cimento 43, 345 (1985)
J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964)
J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966)
J. S. Bell, Introduction to the hidden variable question, Proceedings of the International School of Physics 'Enrico Fermi', Course IL, Foundations of Quantum Mechanics (1971) 171-81
J. S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique, Colloque C2, suppl. au numero 3, Tome 42 (1981) pp C2 41-61
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987) [A collection of Bell's papers, including all of the above.]
J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884 (1969), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Physical Review D, 10, 526-35 (1974)
J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics 41, 1881 (1978)
S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)
E. S. Fry, T. Walther and S. Li, Proposal for a loophole-free test of the Bell inequalities, Phys. Rev. A 52, 4381 (1995)
E. S. Fry, and T. Walther, Atom based tests of the Bell Inequalities - the legacy of John Bell continues, pp 103-117 of Quantum [Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.) (Springer, Berlin-Heidelberg-New York, 2002)
R. B. Griffiths, Consistent Quantum Theory', Cambridge University Press (2002).
L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71 (11) 1665-1668 (1993)
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000)
P. Pearle, Hidden-Variable Example Based upon Data Rejection, Physical Review D 2, 1418-25 (1970)
M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987)
R. García-Patrón Sánchez, J. Fiurácek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier, Proposal for a Loophole-Free Bell Test Using Homodyne Detection, Phys. Rev. Lett. 93, 130409 (2004)
B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.

External Links

An explanation of Bell's Theorem, based on N. D. Mermin's article, "Bringing Home the Atomic World: Quantum Mysteries for Anybody," Am. J. of Phys. 49 (10), 940 (October 1981)
A. Shimony, Bell's Theorem, (2004). [This includes a useful list of references, including general reading.]

Categories: Physics | Quantum information science | Quantum mechanics | Theorems

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