Optimum phase space probabilities from quantum tomography

Abstract

We determine a positive normalised phase space probability distribution P with minimum mean square fractional deviation from the Wigner distribution W. The minimum deviation, an invariant under phase space rotations, is a quantitative measure of the quantumness of the state. The positive distribution closest to W will be useful in quantum mechanics and in time frequency analysis. The position-momentum correlations given by the distribution can be tested experimentally in quantum optics.