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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Bell's inequality and operators' noncommutativity

M. Revzen, M. V. Lokajíček, A. Mann

Published Jun 1, 1997• 11 citations
Publisher

Abstract

Relations between Bell's inequality and noncommutativity of operators are discussed via the four operators involved in the Clauser et al inequality. The case of all operators commuting (i.e. the six commutators vanish) and the case of three out of the four operators mutually commuting (i.e. five commutators vanish) is shown to abide by the inequality. In the latter case a novel insight is unravelled. The Bell quantum bound is obeyed for the case when four commutators vanish. The probabilistic upper limit of the inequality is reviewed and shown to be 4. In any theory based on Hilbert space, the upper limit is .

DOI
10.1088/1355-5111/9/3/016

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Relations between Bell's inequality and noncommutativity of operators are discussed via the four operators involved in the Clauser et al inequality. The case of all operators commuting (i.e. the six commutators vanish) and the case of three out of the four operators mutually commuting (i.e. five commutators vanish) is shown to abide by the inequality. In the latter case a novel insight is unravelled. The Bell quantum bound is obeyed for the case when four commutators vanish. The probabilistic upper limit of the inequality is reviewed and shown to be 4. In any theory based on Hilbert space, the upper limit is .
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.
In this paper we show a Bell inequality of Clauser-Horne type for three three-dimensional systems (qutrits). Violation of the inequality by quantum mechanics is shown for the case in which each of the three observers measures two non-commuting observables, defined by the so called unbiased symmetric six port beamsplitters, on a maximally entangled state of three qutrits. The strength of the violation of the inequality agrees with the numerical results presented in Kaszlikowski et. al., quant-ph//0202019.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.
In constructing his theorem, Bell assumed that correlation functions among non-commuting variables are the same as those among commuting variables. However, in quantum mechanics, multiple data values exist simultaneously for commuting operations while for non-commuting operations data are conditional on prior outcomes, or may be predicted as alternative outcomes of the non-commuting operations. Given these qualitative differences, there is no reason why correlation functions among non-commuting variables should be the same as those among commuting variables, as assumed by Bell. When data for commuting and noncommuting operations are predicted from quantum mechanics, their correlations are different, and they now satisfy the Bell inequality.
Noncommuting observables cannot be simultaneously measured; however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables. We find that the Quasi-Bell inequalities have a quantum to classical violation as high as 32 on two qubit, higher than conventional Bell inequalities. The result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics and provides an insightful generalization of Bell inequalities.