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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Possible experimental test of local commutativity

Paul A. Benioff, H. Ekstein

Published Jun 15, 1977• 5 citations
Publisher

Abstract

A direct experimental test for local commutativity (i.e., $[A, B]=0$ if $A$ and $B$ are associated with mutually spacelike regions) is proposed. It consists of a comparison of the distribution of outcomes of $A$ measurements in the two cases where $A$ alone is measured (no measurement apparatus for $B$ present) and where both are measured on the same sample, in short, a test of the compatibility of the two observation procedures for $A$ and $B$. The implication compatibility \ensuremath{\rightarrow} commutativity (substantially proved by von Neumann, but questioned by Park and Margenau) is proved in a stronger form, with close attention paid to the distinction between different procedures corresponding to a given self-adjoint operator.

DOI
10.1103/physrevd.15.3563

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A direct experimental test for local commutativity (i.e., [A,B]=0[A, B]=0[A,B]=0 if AAA and BBB are associated with mutually spacelike regions) is proposed. It consists of a comparison of the distribution of outcomes of AAA measurements in the two cases where AAA alone is measured (no measurement apparatus for BBB present) and where both are measured on the same sample, in short, a test of the compatibility of the two observation procedures for AAA and BBB. The implication compatibility \ensuremath{\rightarrow} commutativity (substantially proved by von Neumann, but questioned by Park and Margenau) is proved in a stronger form, with close attention paid to the distinction between different procedures corresponding to a given self-adjoint operator.
This thesis addresses two forms of quantum uncertainty. In part I, we focus on preparation uncertainty, an expression of the fact that there are sets of observables for which the induced probability distributions are not simultaneously sharp in any state. We exactly characterise the preparation uncertainty regions for several finite dimensional case studies, including a new derivation of the preparation uncertainty region for the Pauli observables of qubits, and two qutrit case studies which have not previously been addressed in the literature. We also consider the variance based preparation uncertainty for position and momentum observables for the well known “particle in a box” system. We see that the appropriate momentum observable is not given by the spectral measure of a self-adjoint operator, although the position observable is. The box system lacks the phase-space symmetry used to determine the free particle and particle on a ring systems so determining the box uncertainty region is rather more difficult than in these cases. We give upper and lower bounds on the boundary of the uncertainty region, and show that our upper bound is exact in an interval. In part II we turn our attention to measurement uncertainty, exploring the space of compatible joint approximations to incompatible target observables. We prove a general theorem, which shows that, for a broad class of figures of merit, the optimal compatible approximations to covariant targets are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for covariant observables, since the space of covariant compatible approximations is smaller than the space of all compatible approximations. We employ this theorem to derive measurement uncertainty regions for three mutually orthogonal Pauli observables, and for the quantum Fourier pair acting in any finite dimension.
Measurement incompatibility and the negativity of quasi-probability distributions both demonstrate the signature of nonclassicality. However, their relations largely remained qualitative, and no explicit operational connection has been established. We provide an operational link between measurement incompatibility and the Margenau-Hill(MH) quasi-probability distribution associated with two dichotomic observables. { We first derive the joint measurability (measurement compatibility) condition for any pair of dichotomic observables in arbitrary finite dimension ddd.} We then rigorously prove that for any pair of unsharp dichotomic measurements in dimension ddd, the positivity of the MH distribution is equivalent to joint measurability \emph{i.e.}, the MH distribution is positive \emph{if and only if} the measurements are jointly measurable. We further introduce the MH-like quasi-probability distribution for nnn dichotomic unsharp observables in arbitrary finite dimension and derive a sufficient condition of joint-measurability when the observables are mutually anticommuting. Finally, we propose an interferometric setup, inspired by quantum-switch architectures, that directly reconstructs MH quasi-probability which in turn provides a test of incompatibility.
(1) the author has outlined several of the more important interpretations of measurement in quantum mechanics and discussed the problems arising from them. Particular attention was paid to the work of Bohr, Heisenberg and von Neumann and a tentative proposal was made for a possible interpretation which would mitigate some of the problems and dilemmas. This interpretation was essentially that proposed by Margenau (2) in terms of latent variables. He defines measurement to be any operation with physical apparatus which results in a number, including in this, of course, yes and no answers as well as conventional numerical answers. Further he makes the distinction between the preparation of a state and the measurement of the state (3). A preparation puts the quantum system into a particular state whereas the measurement frequently destroys the state in question. This paper presents several criteria to help in the evaluation of the alternatives presented in the first paper and carries further the suggested interpretation mentioned at the end of that paper.
The study of hidden-variable interpretations of quantum mechanics was transformed by two papers of Bell. In his 1966 paper in Reviews of Modern Physics [1] he analysed the argument of von Neumann [2] and the more powerful arguments of Jauch and Piron [3] and of Gleason [4], all leading to similar conclusions about the nonassignability of simultaneous values to all the observables of a nontrivial quantum system; and he showed that these arguments do not exclude what may be called “contextualistic” hidden-variable theories, in which the value of an observable O is allowed to depend not only upon the hidden state λ, but also upon the set C of compatible observables measured along with O . In Bell's 1964 paper in Physics [5] (written later than the other, in spite of the earlier publication date) he gave a new kind of argument against a hidden-variable interpretation of quantum mechanics, by showing that no hidden-variables theory which satisfies a certain condition of locality can agree with all the statistical predictions of quantum mechanics. The main purpose of this lecture is to discuss an experiment, designed by Clauser, Horne, Holt and myself [6], which is based on Bell's proof of the incompatibility of quantum mechanics and local hidden-variable theories. Before doing this, however, I wish to make several comments.