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arXiv:1911.08693

Wigner functions in the Higher-Spin Einstein-Podolsky-Rosen-Bohm Experiment

Anupam Garg

Published Nov 20, 2019• 0 citations
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Abstract

The spin-$j$ extension of Bohm's version of the Einstein-Podolsky-Rosen experiment is is analysed in terms of the Wigner function when the two spins are in a singlet state. This function is calculated for all $j$, and it is shown that just as Bell inequalities are violated with undiminished range and magnitude for arbitarirly large $j$, this function does not become less negative. On the contrary, the oscillations between positive and negative values grow both in frequency and amplitude. It is argued that this is an alternative way to grasp the approach to classical behavior with increasing quantum number.

DOI
10.48550/arxiv.1911.08693

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The spin-jjj extension of Bohm's version of the Einstein-Podolsky-Rosen experiment is is analysed in terms of the Wigner function when the two spins are in a singlet state. This function is calculated for all jjj, and it is shown that just as Bell inequalities are violated with undiminished range and magnitude for arbitarirly large jjj, this function does not become less negative. On the contrary, the oscillations between positive and negative values grow both in frequency and amplitude. It is argued that this is an alternative way to grasp the approach to classical behavior with increasing quantum number.
We consider the Einstein-Podolsky-Rosen correlation between two spins with arbitrary j values using the phase-space description of the angular-momentum operators [G. S. Agarwal, Phys. Rev. A 24, 2889 (1981)]. The phase-space formulation provides insight into hidden-variable theories and enables us to understand why Bell's inequalities are violated. We use the analogs of the Wigner function and the P function for the analysis of the correlations. We show that the Wigner function for the singlet state becomes positive in the limit j\ensuremath{\rightarrow}\ensuremath{\infty}. We also show the extent to which Bell's inequalities can be violated for higher spins and present physical quantities, the correlations between which would violate Bell's inequalities.
Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-sss particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of NNN axes are incompatible with local realism for any set of distinct coplanar axes when N=3N=3N=3, and for any set of distinct axes whatever when N=4N=4N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.
We calculate the joint probabilities and the correlation function in Einstein-Podolsky-Rosen--type experiments with a massive vector boson in the framework of quantum field theory. We report on the strange behavior of the correlation function (and the probabilities)---the correlation function, which in the relativistic case still depends on the particle momenta, for some fixed configurations has local extrema. We also show that relativistic spin-1 particles violate some Bell inequalities more than nonrelativistic ones and that the degree of violation of the Bell inequality is momentum dependent.
A Bell inequality violation allowed by the two-mode squeezed state, whose Wigner function is nonnegative, is shown to hold only for correlations among dynamical variables that cannot be interpreted via a local hidden variable theory. Explicit calculations and interpretation are given for Bell's suggestion that the EPR (Einstein, Podolsky, and Rosen) state will not allow violation of Bell's inequality, in conjunction with its Wigner representative being nonnegative. It is argued that Bell's theorem disallowing the violation of Bell's inequality within a local hidden-variable theory depends on the dynamical variables having a definite value---assigned by the local hidden variables---even when they cannot be simultaneously measured. The analysis leads us to conclude that Bell's inequality violation is to be associated with endowing these definite values to the dynamical variables, and not with their locality attributes.
The spin-jjj Einstein-Podolsky-Rosen experiment is studied with a view to understanding how classical behaviour emerges as j→∞j \to \inftyj→∞. It is proposed that it is necessary to include detector error, which if it is be to viewed as an essential aspect of the emergence of classicality, should be both minimal, i.e., no more than necessary to wash out quantum mechanical behaviour, and agnostic, by which is meant that one should be able to ascribe it to error in the preparation of the state just as well as to the detector.Errors in the state preparation are discussed via the spin Wigner function. An agnostic error protocol is described which appears to be minimal.
We show that configurations exist in which the correlation functions and the degree of violation of Bell-type inequalities in the relativistic Einstein-Podolsky-Rosen (EPR) experiment have local extrema for some values of the velocities of the EPR particles. Moreover, this strange behavior can be observed for both discussed relativistic spin operators and for spin-1∕21∕21∕2 as well as spin-1 particles.