How to check whether a given $W(x,p)$ represents a Wigner function of a physical state?

Question

For simplicity let us consider one-dimensional quantum-mechanical systems only. Given any state $\rho\in\mathcal{B}(\mathcal{H})$ and its Wigner function $W_\rho(x,p)$, there are several properties it always satisfies,

• it is real-valued and bounded in absolute value by $2/\hbar$ from above,
• its marginals are well-behaved probability distributions,
• its overlap $\int W_\rho(x,p) W_\sigma(x,p) \mathrm{d}x\mathrm{d}p$ with the Wigner function of any other state $\sigma$ is nonnegative.

Given an arbitrary function $W(x,p)$, we can reconstruct a state $\rho$ it corresponds to iff all of these conditions are met using the Weyl transform. If some properties are not satisfied, the resulting operator will be unphysical—non-Hermitian, unnormalized or non-positive. The question is how to check the conditions prior to the calculation.

The first two are usually fairly easy to check but the latter involves going through all states of the system. Even if we restrict ourselves to pure states, this still iterates over all rays in an infinite-dimensional linear vector space and is as such usually done finding a proof tailored to the specific case. Is there a simpler (algorithmic) condition which can replace the third bullet above and implies its validity?

This post imported from StackExchange Physics at 2014-09-02 20:42 (UCT), posted by SE-user Vašek Potoček

Comments

This is very difficult to check explicitly, since in the Wigner description, quantum positivity is not a natural condition. One usually proceeds in the opposite direction: one knows that a state is positive, then calculates its Wigner transform, and can be sure that the required conditions are met. 

Answer 1

There is another formulation, not algorithmic, but the exact analog of positiv definiteness in Bochner's theorem for Fourier transforms of probability measure. This is called $\omega$-positivity (or $\omega$-positive definiteness, or $\omega$-non negative definiteness). The (inverse) Fourier transform of the Wigner function has to be $\omega$-positive, a condition like positive definiteness, but with an additional function called $\omega$, see the paper

Robin L. Hudson, Quantum Bochner theorems and incompatible observables,  Kybernetika (2010) Volume 46, Issue: 6, page 1061-1068, ISSN: 0023-5954, http://eudml.org/doc/196490

and the references therein.