Bounds on Integrals of the Wigner Function

The integral of the Wigner function over a subregion of the phasespace of a quantum system may be less than zero or greater than one. It is shown that for systems with one degree of freedom, the problem of determining the best possible upper and lower bounds on such an integral, over all possible states, reduces to the problem of finding the greatest and least eigenvalues of an hermitian operator corresponding to the subregion. The problem is solved exactly in the case of an arbitrary elliptical region. These bounds provide checks on experimentally measured quasiprobability distributions. The Wigner function has been much studied since its introduction [1], not only in the context of quantum physics [2], but also in signal processing [3]. For a quantum system in a pure state, the Wigner function carries the