Where the Quantum-Memory Advantage Lives: Copy Complexity of the Harmonics of a Shifted Self-Overlap

This preprint is the second part of a series on Fourier harmonics of shifted overlaps under a U(1) symmetry. The first part (Fourier sectors of shifted overlaps under a compact abelian symmetry, doi:10.5281/zenodo.23137889) introduced the framework: the overlap of two states, one shifted by a group element, decomposes into a finite Fourier series whose coefficients are positive semi-definite sector kernels, read exactly from 2n+1 shifts. That paper made no advantage claim. This paper determines the copy complexity of the diagonal of those kernels for an unknown n-qubit mixed state: the harmonics κ_m(ρ) = ‖A_m ρ‖₂² of the self-overlap Tr(ρ e^{iΔN} ρ e^{−iΔN}), with N the number operator and A_m the projection onto relative charge m. For a reduced state graded by its subsystem charge, these are the replica-two charged moment of the entanglement asymmetry and, normalised, the sector weights of symmetry-resolved operator entanglement. The setting is the model of learning with and without quantum memory of Chen, Cotler, Huang and Li. The results are unconditional. With a quantum memory of one copy, swap tests between a copy and a shifted copy at 2n+1 shifts return every harmonic within ε from O(ε⁻² log(n/δ) + n) pairs of copies. Without quantum memory, and whatever the classical memory and adaptivity, a single harmonic needs Ω(max{ε⁻², √D_{n,m}/ε}) copies, where D_{n,m} = C(n, ⌊(n−m)/2⌋) is the dimension of the charge sectors at distance m; O((2n+1)(ε⁻² + 2^{n/2}ε⁻¹) ln((2n+1)/δ)) single-copy measurements suffice. The separation between learning with and without quantum memory therefore survives symmetry resolution, with its lower bound fixed by the sector dimension rather than by 2ⁿ. The paper locates the boundaries of this separation. It is absent for pure states, where charge statistics suffice, and for n−m bounded, where polynomially many single-copy measurements suffice. It survives unital phase-covariant noise, including charge dephasing of any strength and depolarising noise up to a fixed rate. The bounded-memory bounds of Gong, Haferkamp, Ye and Zhang transfer to each harmonic. A dimension cap shows how far reductions affine in the purity can reach. A final corollary carries the lower bound to the normalised sector weights and to the Rényi-2 entanglement asymmetry: at constant error, estimating the entanglement asymmetry of an n-qubit mixed state from single copies costs Ω(2^{n/2}/n^{1/4}) copies, while two swap tests between pairs of copies return it from O(1) pairs for states of bounded purity. The lower-bound technique is that of Chen, Cotler, Huang and Li (purity testing) and Anshu, Landau and Liu (distributed inner-product estimation), applied through an embedding of a 2^k-dimensional problem across two charge sectors. The purity-testing bound is reproved in an appendix with an explicit constant. No advantage is claimed for states given with a classical description, for which every harmonic is classically computable in polynomial time. The repository contains a numpy implementation, 65 tests tied to the numbered statements, and the script that regenerates every figure and number of the paper from a single seed. Keywords: quantum learning theory; copy complexity; quantum memory; symmetry-resolved entanglement; entanglement asymmetry; modes of asymmetry; swap test; purity estimation; U(1) symmetry