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

Quantum Bochner's theorem for phase spaces built on projective representations