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Physics of Atomic Nuclei

Investigation of properties of time-dependent bell inequalities in Wigner’s form for nonstationary and open quantum systems

N. Nikitin, V.P. Sotnikov, K. S. Toms

Published Oct 1, 2015• 2 citations
Publisher

Abstract

A radically new class of Bell inequalities in Wigner’s form was obtained on the basis of Kolmorov’s axiomatization of probability theory and the hypothesis of locality. These inequalities take explicitly into account the dependence on time (time-dependent Bell inequalities in Wigner’s form). By using these inequalities, one can propose a means for experimentally testing Bohr’ complementarity principle in the relativistic region. The inequalities in question open broad possibilities for studying correlations of nonrelativistic and relativistic quantum systems in external fields. The violation of the time-dependent inequalities in quantum mechanics was studied by considering the behavior of a pair of anticorrelated spins in a constant external magnetic field and oscillations of neutral pseudoscalar mesons. The decay of a pseudoscalar particle to a fermion–antifermion pair is considered within quantum field theory. In order to test experimentally the inequalities proposed in the present study, it is not necessary to perform dedicated noninvasive measurements required in the Leggett–Garg approach, for example.

DOI
10.1134/s1063778815070108

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We propose time-dependent Bell inequalities in a Wigner form, which expand the possibilities of experimental verification of Bohr’s principle of complementarity in the relativistic domain and for nonstationary quantum-mechanical systems. Derivation of the proposed inequalities is based entirely on the Kolmogorov axiomatics of probability theory and the locality hypothesis. Violation of the obtained inequalities is considered in the framework of quantum theory based on the example of oscillations of neutral B mesons.
Recently a new class of time-dependent Bell inequalities in Wigner form was introduced. The structure of the inequalities allows experimental studies of quantum and open quantum systems in external fields. In this paper, we study the properties of the time-dependent Wigner inequalities using the time evolution of neutral pseudoscalar mesons with CPCPCP-violation effects taken into account. It is shown that it is always possible to find a range of parameters to test for violation in an experimentally accessible area. The effect of the relaxation of the inequalities for large time scales is demonstrated.
In this paper we propose a time-independent equality and time-dependent inequality, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well-known Greenberger-Horne-Zeilinger (GHZ) equality and its variants, because violation of the proposed equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $|{\mathrm{\ensuremath{\Psi}}}^{\ensuremath{-}}\ensuremath{\rangle}$, while a test of the GHZ equality requires at least three quantum systems in a special state $|{\mathrm{\ensuremath{\Psi}}}^{\mathrm{GHZ}}\ensuremath{\rangle}$. The obtained inequality differs from Bell's, Wigner's, and Leggett-Garg inequalities, because it deals with spin s=1/2s=1/2s=1/2 projections onto only two nonparallel directions at two different moments of time, while a test of the Bell and Wigner inequalities requires at least three nonparallel directions, and a test of the Leggett-Garg inequalities requires at least three distinct moments of time. Hence, the proposed inequality seems to open an additional experimental possibility to avoid the ``contextuality loophole.'' Violation of the proposed equality and inequality is illustrated with the behavior of a pair of anticorrelated spins in an external magnetic field and also with the oscillations of flavor-entangled pairs of neutral pseudoscalar mesons.
We present a relativistic generalization of the Wigner inequality for the scalar and pseudoscalar particles decaying to two particles with spin (fermions and photons.) We consider Wigner's inequality with the full spin anticorrelation (with the nonrelativistic analog), as well as the case with the full spin correlation. The latter case may be obtained by a special choice of the plane of measurement of the spin projections on the direction of propagation of fermions. The possibility for relativistic testing of Bohr's complementarity principle is shown.