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Journal of Mathematical Physics

Joint distributions, quantum correlations, and commuting observables

Arthur I. Fine

Published Jul 1, 1982• 337 citations
Publisher

Abstract

We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.

DOI
10.1063/1.525514

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Cambridge University Press eBooks•Nov 13, 1996•0 citations
We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
It is shown that the following statements about a quantum correlation experiment are mutually equivalent. (1) There is a deterministic hidden-variables model for the experiment. (2) There is a factorizable, stochastic model. (3) There is one joint distribution for all observables of the experiment, returning the experimental probabilities. (4) There are well-defined, compatible joint distributions for all pairs and triples of commuting and noncommuting observables. (5) The Bell inequalities hold.
Joint quantum measurements of noncommuting observables are possible, if one accepts an increase in the measured variances. A necessary condition for a joint measurement to be possible is that a joint probability distribution exists for the measurement. This fact suggests that there may be a link with Bell inequalities, as these will be satisfied if and only if a joint probability distribution for all involved observables exists. We investigate the connections between Bell inequalities and conditions for joint quantum measurements to be possible. Mermin's inequality for the three-particle Greenberger-Horne-Zeilinger state turns out to be equivalent to the condition for a joint measurement on two out of the three quantum systems to exist. Gisin's Bell inequality for three coplanar measurement directions, meanwhile, is shown to be less strict than the condition for the corresponding joint measurement.
Under a standard set of assumptions for a hidden-variable model for quantum events we show that all observables must commute simultaneously. This seems to be an ultimate statement about the inapplicability of the usual hidden-variable model for quantum events. And, despite Bell's complaint that a key condition of von Neumann's was quite unrealistic, we show that these conditions, under which von Neumann produced the first no-go proof, are entirely equivalent to those introduced by Bell and Kochen and Specker. As these conditions are also equivalent to those under which the Bell-Clauster-Horne inequalities are derived, we see that the experimental violations of the inequalities demonstrate only that quantum observables do not commute.
We derive inequalities for n-partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than 1∕2 is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An n-partite uncorrelated state violates the n-partite inequality derived here by an amount that grows exponentially with n.
INTRODUCTION Fine (1982) proved that Bell's (1964) inequalities are sufficient, as well as necessary, to guarantee the existence of a joint probability distribution of the random variables in question, e.g., those representing certain spin correlation experiments. Garg and Mermin (1982) gave an example of eight random variables satisfying Bell's inequalities but such that no joint distribution exists. The purpose of this paper is to extend Bell's inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that, for N > 4, many new elementary inequalities beyond those of Bell must be satisfied by any hidden-variable theory. Since these additional inequalities are violated by quantum mechanical predictions for appropriate choice of measurement arrangements, they serve to increase the conceptual distance between what may be called Einstein locality, after the EPR paradox, and quantum mechanics. To give a concrete sense of the nature of our results, we exhibit one of the new inequalities violated by Garg and Mermin's example. For theoretical purposes it will suffice to introduce Bell-type random variables X 1 ,…, X N , N ≥ 4, having possible values ±1, with means E ( X i ) = 0 and with what we term Bell covariances E ( X i X j ) relative to the index i 0 , with 1 ≤ i ≤ i 0 , N , 2 ≤ i 0 N – 2. Notice that, when N = 4, we must, as is familiar, have i 0 = 2.