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Axioms

Classical Probability and Quantum Outcomes

James D. Malley

Published May 26, 2014• 1 citation
Publisher

Abstract

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.

DOI
10.3390/axioms3020244

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There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.
We discuss quantum conditional probability and its applications to deterministic hidden-variable models. We derive empirical tests corresponding to mathematical no-go proofs, providing rigorous statistical tests based on experimental outcomes. Evidently, it now possible to examine the statistical power of the empirical tests, and place confidence intervals on the parameters that precisely measure the departure of hidden-variable models from quantum experimental outcomes. Moreover, reinterpretation of well-known results in the light of quantum conditional probability provides other experimental demonstrations and no-go proofs: outcomes for the familiar Young double-slit experiment show that there are no deterministic hidden-variable models of the type considered by Kochen and Specker [J. Math. Mech. 17, 59 (1967)] or Bell [Rev. Mod. Phys. 38, 447 (1966)].
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.
In this talk I present a simple derivation of an old result of Kochen and Specker, which is apparently unrelated to the famous work of Bell on hidden variables, but is presumably equally important. Kochen and Specker showed in 1967 that quantum mechanics cannot be embedded into a classical stochastic theory, provided the quantum theoretical probability distributions are reproduced and one additional highly desirable property is satisfied. This showed in a striking manner what were the difficulties in implementing the Einstein programme of a `complete' version of quantum mechanics.
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves.
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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.