  Bochner's theorem gives the necessary and sufficient conditions on a function
such that its Fourier transform corresponds to a true probability density
function. In the Wigner phase space picture, quantum Bochner's theorem gives
the necessary and sufficient conditions on a function such that it is a quantum
characteristic function of a valid (and possibly mixed) quantum state and such
that its Fourier transform is a true probability density. We extend this
theorem to discrete phase space representations which possess enough symmetry.
More precisely, we show that discrete phase space representations that are
built on projective unitary representations of abelian groups, with a slight
restriction on admissible 2-cocycles, enable a quantum Bochner's theorem.
