Characterizing quantum resourcefulness via group-Fourier decompositions

TL;DR

This paper introduces a group Fourier decomposition framework to characterize quantum resourcefulness in pure states, revealing universal patterns across various resource theories and linking mathematical properties to operational measures.

Contribution

The work develops a general GFD-based approach to quantify and analyze resourcefulness in quantum states, connecting harmonic analysis with quantum resource theories.

Findings

Low-resource states are supported in small-dimensional irreps.

High-resource states occupy higher-dimensional irreps.

GFD purities serve as resourcefulness witnesses and relate to state compressibility.

Abstract

In this work we present a general framework for studying the resourcefulness in pure states for quantum resource theories (QRTs) whose free operations arise from the unitary representation of a group. We argue that the group Fourier decompositions (GFDs) of a state, i.e., its projection onto the irreducible representations (irreps) of the Hilbert space, operator space, and tensor products thereof, constitute fingerprints of resourcefulness and complexity. By focusing on the norm of the irrep projections, dubbed GFD purities, we find that low-resource states live in the small dimensional irreps of operator space, whereas high-resource states have support in more, and higher dimensional ones. Such behavior not only resembles that appearing in classical harmonic analysis, but is also universal across the QRTs of entanglement, fermionic Gaussianity, spin coherence, and Clifford stabilizerness. To finish, we show that GFD purities carry operational meaning as they lead to resourcefulness witnesses as well as to notions of state compressibility.