Exact quantum correlations of conjugate variables from joint quadrature measurements

Abstract

We demonstrate that for two canonically conjugate operators q̂, p̂, the global correlation 〈q̂p̂+p̂q̂〉−2〈q̂〉〈p̂〉, and the local correlations 〈q̂〉(p)−〈q̂〉 and 〈p̂〉(q)−〈p̂〉 can be measured exactly by Von Neumann–Arthurs–Kelly joint quadrature measurements. Here 〈p̂〉(q) and 〈q̂〉(p) denote the conditional expectation values of momentum at a given position, and position at a given momentum respectively. These correlations provide a sensitive experimental test of quantum phase space probabilities quite distinct from the probability densities of q, p. E.g. for EPR states, and entangled generalized coherent states, phase space probabilities which reproduce the correct position and momentum probability densities have to be modified to reproduce these correlations as well.