Complete conditions for legitimate Wigner distributions

Given a real-valued phase-space function, it is a nontrivial task to determine whether it corresponds to a Wigner distribution for a physically acceptable quantum state. This topic has been of fundamental interest for a long time, and in a modern application, it can be related to the problem of entanglement detection for multimode cases. In this paper, we present a hierarchy of complete conditions for a physically realizable Wigner distribution. Our derivation is based on the normally ordered expansion, in terms of annihilation and creation operators, of the quasidensity operator corresponding to the phase-space function in question. As a by-product, it is shown that the phase-space distributions with elliptical symmetry can be readily diagonalized in our representation, facilitating the test of physical realizability. We also illustrate how the current formulation can be connected to the detection of bipartite entanglement for continuous variables.