Bell's theorem

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Template:Mergefrom Template:Technical Bell's theorem refers to a class of correlation inequalities that hold under local realism but do not apply under quantum mechanics (QM). The Theorem is named after John Bell, whose ground-breaking mid-1960's papers examined both von Neumann's proof of the non-existence of hidden variables (1932) and the EPR paradox (1935) in greater detail. Bell's most famous paper is entitled On the Einstein Podolsky Rosen Paradox (1964).

After EPR, the scientific community had been left in the uncomfortable position that QM appeared accurate but incomplete. In this view, local hidden variables existed but were not described by QM (and thus QM must be incomplete). However, Bell showed that seemingly "reasonable" assumptions within EPR about reality led to a contradiction with the predictions of quantum mechanics. The assumptions were that a) no effect can propagate faster than the speed of light (this is the requirement of locality); and b) that the likelihood of any permutation of the hypothesized hidden variables occurring was between 0% and 100%. Bell saw that this was not always the case when considering spin polarization components with entangled particles at specific angles. Subsequent experimental tests of Bell's Inequalities are consistent with the predictions of QM. However, they are inconsistent with at least one of the assumptions of local reality. Therefore, either a) or b) is incorrect.

Bell considered a hypothetical 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 XA or YA, and Bob can choose between detector settings to measure either XB or YB. 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):

<math> (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, <math>

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

Correlation

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

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

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

<math> \mathbf{C}(X,Y) = \operatorname{E}(X Y) <math>

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

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

almost surely. Equation (2) makes sense for any sequence of measured values Xn, Yn. 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).

Formulation

We can now state a form of Bell's theorem, although we have not presented here a mathematical theory in which it follows deductively. For approaches that explicitly formalize local realism, see the page on the CHSH inequality, where two derivations are given, including one by Bell (Bell, 1971). See also Shimony's Stanford Encyclopedia of Philosophy article referenced below where several approaches to formalization of local realism are discussed; see also properties of interpretations. The approach followed in this article closely parallels that given in (Peres, 1993) and (Nielsen and Chuang, 2000), both of which are standard references in quantum theory.

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 XAn, YAn, XBn, YB n, 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

<math> \mathbf{C}(X_A, X_B), \mathbf{C}(X_A,Y_B), \mathbf{C}(Y_A, X_B), \mathbf{C}(Y_A,Y_B) <math>

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

<math> X_B^n + Y_B^n, \quad X_B^n - Y_B^n <math>

is zero regardless of which of the variables XA or YA is measured by Alice. Thus

<math> \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, <math>

since each summand can be regrouped as

<math> 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. <math>

Thus

<math> \mathbf{C}(X_A, X_B)+ \mathbf{C}(X_A,Y_B)+ \mathbf{C}(Y_A, X_B) - \mathbf{C}(Y_A,Y_B) = <math>
<math> = \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 <math>
<math> = \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) <math>
<math> \leq \lim_{N \rightarrow \infty} \frac{1}{N} \ 2 N = 2. \quad <math>

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:

  1. A measures XAn, B measures XBn,
  2. A measures XAn, B measures YBn,
  3. A measures YAn, B measures XBn,
  4. A measures YAn, B measures YBn.

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

Remark 2. The validity of the correlation inequality (1) still holds if the variables XA, YA, XB, YB 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:

<math> X_A \ X_B + X_A \ Y_B + Y_A \ X_B - Y_A \ Y_B = <math>
<math> = X_A (X_B + Y_B) +Y_A (X_B - Y_B) \quad <math>
<math> \leq \bigg|X_A (X_B + Y_B) +Y_A (X_B - Y_B) \bigg| \quad <math>
<math> \leq \bigg|X_A (X_B + Y_B)\bigg| +\bigg|Y_A (X_B - Y_B)\bigg| <math>
<math> \leq |X_B + Y_B| +| X_B - Y_B| \leq 2.<math>

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

<math> X_B \geq Y_B \geq 0. <math>

In that case

<math> |X_B + Y_B| +| X_B - Y_B| =X_B + Y_B + X_B - Y_B = 2 X_B \leq 2. <math>

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

<math> \|\operatorname{E}_X(\lambda) \phi\|^2 <math>

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

<math> \|\operatorname{E}_X(\lambda) \phi\|^{-1} \operatorname{E}_X(\lambda) \phi.<math>

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

<math> \langle X Y \rangle = \langle X Y \psi \mid \psi \rangle. <math>

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:

<math> S_x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} <math>
<math> S_z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. <math>

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

<math> \left|+x\right\rang, \quad \left|-x\right\rang. <math>

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

<math>

\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). <math>

Now let us apply the CHSH formalism to the observables:

<math> X_A = S_z \otimes I <math>
<math> Y_A = S_x \otimes I <math>
<math> X_B = - \frac{1}{\sqrt{2}} \ I \otimes (S_z + S_x) <math>
<math> Y_B = \frac{1}{\sqrt{2}} \ I \otimes (S_z - S_x). <math>
The operators XB, YB correspond to B's spin measurements along skew directions in the xz-plane.
Missing image
Bells-thm.png
Illustration of Bell test for spin 1/2 particles. Source produces spin singlet pair, one particle sent to Alice another to Bob. Each performs one of the two spin measurements.
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
<math> \langle X_A X_B \rangle = \langle X_A Y_B \rangle =\langle Y_A X_B \rangle = \frac{1}{\sqrt{2}}, <math>

and

<math> \langle Y_A Y_B \rangle = - \frac{1}{\sqrt{2}}. <math>

so that

<math> \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. <math>

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 <math>2 \sqrt{2}<math> 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".
Missing image
Bell-test-photon-analyer.png
Scheme of a "two-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (++, --, +- and -+) counted by the coincidence monitor.

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."

See also

Further reading

The following are intended for general audiences.

  • Amir D. Aczel, Entanglement: The greatest mystery in physics (Four Walls Eight Windows, New York, 2001).
  • A. Afriat and F. Selleri, The Einstein, Podolsky and Rosen Paradox (Plenum Press, New York and London, 1999)
  • J. Baggott, The Meaning of Quantum Theory (Oxford University Press, 1992)
  • N. David Mermin, "Is the moon there when nobody looks? Reality and the quantum theory", in Physics Today, April 1985, pp. 38-47.
  • D. Wick, The infamous boundary: seven decades of controversy in quantum physics (Birkhauser, Boston 1995)

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)
  • A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
  • M. Redhead, Incompleteness, Nonlocality and Realism, Clarendon Press (1987)
  • B. C. van Frassen, Quantum Mechanics, Clarendon Press, 1991.

External links

pt:teorema de Bell

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