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

TL;DR

This paper extends Bochner's theorem to discrete quantum phase spaces constructed from projective unitary representations of abelian groups, providing necessary and sufficient conditions for quantum states in these frameworks.

Contribution

It introduces a quantum Bochner's theorem for discrete phase spaces built on projective representations, broadening the scope of phase space quantum state characterization.

Findings

Extended Bochner's theorem to discrete phase spaces with symmetry

Identified conditions on 2-cocycles for the theorem to hold

Provided a framework for quantum state characterization in discrete settings

Abstract

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.