Description
In this talk we present a framework for studying resourcefulness in quantum resource theories (QRTs) whose free operations are generated by a unitary representation of a group. Our central tool is the Group Fourier decomposition (GFD)—the projection of a state onto irreducible representations—whose component norms provide compact “fingerprints” that certify and stratify resourcefulness, yielding witnesses across entanglement, coherence, stabilizer-ness, fermionic Gaussianity, and more. We close by showing how unitary free operations in Lie-algebraic QRTs can be promoted to resource-nonincreasing channels that provably map free states to free states. This extends the Local Unitary→SLOCC transition in entanglement and yields new free operations in QRTs such as that of fermionic Gaussianity.
