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Physical Review A

Quantum conditional probability and hidden-variables models

James D. Malley

Published Aug 1, 1998• 7 citations
Publisher

Abstract

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)].

DOI
10.1103/physreva.58.812

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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)].
Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. We start with a quantum-mechanical model and describe various properties that can be asked of a hidden-variable model. We present six such properties and a Venn diagram of how they are related. With two existence theorems and three no-go theorems (EPR, Bell, and Kochen-Specker), we show which properties of empirically equivalent hidden-variable models are possible and which are not. Formally, our treatment relies only on classical probability models, and physical phenomena are used only to motivate which models to choose.
A recent proposal to experimentally test quantum mechanics against noncontextual hidden-variable theories [Phys. Rev. Lett.80, 1797 (1998)] is shown to be related with the smallest proof of the Kochen–Specker theorem currently known [Phys. Lett.A212, 183 (1996)]. This proof contains eighteen yes-no questions about a four-dimensional physical system, combined in nine mutually incompatible tests. When these tests are considered as tests about a two-part two-state system, then quantum mechanics and noncontextual hidden variables make the same predictions for eight of them, but make different predictions for the ninth. Therefore, this ninth test would allow us to discriminate between quantum mechanics and noncontextual hidden-variable theories in a (gedanken) single run experiment.
A theoretical investigation is made into the difficulties encountered in constructing a deterministic model for quantum mechanics and into the restrictions that can be placed on the form of such a model.The various implications of the known 11 impossi bil ity proofs II are examined.A possible explanation for the non-locality required by Bell's proof is suggested in terms of backwards-in-time causality.The efficacy of the Kochen and Specker proof is brought into doubt by showing that there is a possible way of avoiding its implications in the only known physically realizable situation to which it applies.Also, a new thought experiment is put forward to show that a particle's predetermined momentum and energy values cannot satisfy the laws of momentum and energy conservation without conflicting with the predictions of quantum mechanics.Special attention is paid to a particular class of deterministic models which appear not to have been explicitly studied before.The models considered are those for which the individual outcomes of measurements are not dependent on hidden variables associated with the measuring apparatus and for which the hidden variables of a particle do not need to be randomized after each measurement.Specific models of this type are constructed and examined for the cases of both discrete and continuous spectra of eigenvalues.In developing these models a need is found for a joint distribution for position and momentum.In relation to this point, it is demonstrated that the quantum mechanical wave equations actually point to a particular joint distribution expression, namely the Margenau and Hill distribution.The schemes considered are found to have certain shortcomings and possible steps towards more satisfactory models are suggested.
This paper discusses some tests of Bell-like inequalities not requiring entangled states. The proposed tests are based on consecutive measurements on a single qubit. Available hidden-variable models for a single qubit [see, e.g., J. S. Bell, Rev. Mod. Phys. 38, 447 (1966)] reproduce the predictions of quantum mechanics and hence violate the Bell-like inequalities addressed in this paper. It is shown how this fact is connected with the state ``collapse'' and with its random nature. Thus, it becomes possible to test truly realistic and deterministic hidden-variable models. In this way, it can be shown that a hidden-variable model should entail at least one of the following features: (i) nonlocality, (ii) contextuality, or (iii) discontinuous measurement-dependent probability functions. The last two features are put to the test with the experiments proposed in this paper. A hidden-variable model that is noncontextual and deterministic would be at variance with some predictions of quantum mechanics. Furthermore, the proposed tests are more likely to be loophole-free, as compared to former ones.
Certain concrete "ontological models" for quantum mechanics (models in which measurement outcomes are deterministic and quantum states are equivalent to classical probability distributions over some space of `hidden variables') are examined. The models are generalizations of Kochen and Specker's such model for a single 2-dimensional system - in particular a model for a three dimensional quantum system is considered in detail. Unfortunately, it appears the models do not quite reproduce the quantum mechanical statistics. They do, however, come close to doing so, and in as much as they simply involve probability distributions over the complex projective space they do reproduce pretty much everything else in quantum mechanics. The Kochen-Specker theorem is examined in the light of these models, and the rather mild nature of the manifested contextuality is discussed.