Bell Inequalities in Phase Space and their Violation in Quantum Mechanics

Details Bell Inequalities in Phase Space and their Violation in Quantum Mechanics Bell Inequalities in Phase Space and their Violation in Quantum Mechanics

2002-05-24

arxiv.org/abs/quant-ph/0205157v1

Physics

Quantum Physics

11 pages, latex, no figures

Scientific paper

10.1016/S0375-9601(02)00827-7

We derive ``Bell inequalities'' in four dimensional phase space and prove the following ``three marginal theorem'' for phase space densities $\rho(\overrightarrow{q},\overrightarrow{p})$, thus settling a long standing conjecture : ``there exist quantum states for which more than three of the quantum probability distributions for $(q_1,q_2)$, $(p_1,p_2)$, $(q_1,p_2)$ and $(p_1,q_2)$ cannot be reproduced as marginals of a positive $\rho(\overrightarrow{q},\overrightarrow{p})$''. We also construct the most general positive $\rho(\overrightarrow{q},\overrightarrow{p})$ which reproduces any three of the above quantum probability densities for arbitrary quantum states. This is crucial for the construction of a maximally realistic quantum theory.

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