Bell Inequalities in Phase Space and their Violation in Quantum Mechanics

Abstract

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.

11 pages, latex, no figures

References in corpus (3)

• Proposal for the Measurement of Bell-like Correlations from Continuous Variables
• Marginal distributions in $(2N)$-dimensional phase space and the quantum $(\mathbf{N}+1)$ marginal theorem
• Contextual Deterministic Quantum Mechanics

Cited by in corpus (12)

• Maximal Violation of Bell Inequalities using Continuous Variables Measurements
• Bell inequalities for continuous-variable correlations
• Degaussification of twin-beam and nonlocality in the phase space
• Maximal violation of Bell inequalities by position measurements
• Violation of Bell inequalities in an analogue black hole
• Bell Inequalities in Four Dimensional Phase Space and the Three Marginal Theorem
• Optimum Phase Space Probabilities From Quantum Tomography
• Marginal distributions in $(2N)$-dimensional phase space and the quantum $(\mathbf{N}+1)$ marginal theorem
• Exact Quantum Correlations of Conjugate Variables From Joint Quadrature Measurements
• Joint Probabilities Reproducing Three EPR Experiments On Two Qubits
• Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function
• Binary Homodyne Detection for Observing Quadrature Squeezing in Satellite Links