Group Fourier filtering of quantum resources in quantum phase space

TL;DR

This paper introduces a framework using group Fourier analysis in quantum phase spaces to characterize, filter, and visualize quantum resources, revealing new connections between phase space representations and resource theories.

Contribution

It demonstrates that the entire family of Stratonovich-Weyl quantum phase space representations can be interpreted as group Fourier filters, providing a new signal-processing perspective on quantum resource theories.

Findings

Group Fourier filters can favor low or high-dimensional irreps in quantum states.

Norms of free state Fourier components fully characterize quantum phase space representations.

An s-duality relates phase space spectra of free and resourceful states.

Abstract

Recently, it has been shown that group Fourier analysis of quantum states, i.e., decomposing them into the irreducible representations (irreps) of a symmetry group, enables new ways to characterize their resourcefulness. Given that quantum phase spaces (QPSs) provide an alternative description of quantum systems, and thus of the group's representation, one may wonder how such harmonic analysis changes. In this work we show that for general compact Lie-group quantum resource theories (QRTs), the entire family of Stratonovich-Weyl quantum phase space representations-characterized by the Cahill-Glauber parameter $s$-has a clear resource-theoretic and signal-processing meaning. Specifically, changing $s$ implements a group Fourier filter that can be continuously tuned to favor low-dimensional irreps where free states have most of their support ($s=-1$), leave the spectrum unchanged ($s=0$), or highlight resourceful, high-dimensional irreps ($s=1$). As such, distinct QPSs constitute veritable group Fourier filters for resources. Moreover, we show that the norms of the QRT's free state Fourier components completely characterize all QPSs. Finally, we uncover an $s$-duality relating the phase space spectra of free states and typical (Haar-random) highly resourceful states through a shift in $s$. Overall, our results provide a new interpretation of QPSs and promote them to a signal-processing framework for diagnosing, filtering, and visualizing quantum resources.