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
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
A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.
For two pure states ψ, φ of an n-qubit register, the overlap after a common phase shift, g(θ) = ⟨ψ|e^{iθN}|φ⟩ with N the number operator, is a trigonometric polynomial of degree at most n. When g does not vanish on the circle, its winding number W(ψ, φ), the shift winding of the pair, counts the zeros in the unit disk of the polynomial whose coefficients are the overlaps within each Hamming-weight sector. W is invariant under independent global phases, under common shifts and under every joint unitary that commutes with N. Along a path of pairs it can change only where some common shift makes the two states orthogonal. The sector kernels are blind to W. These kernels are the Fourier coefficients of |g|², measured by swap tests on a shift grid; their diagonal values are the multiple-quantum-coherence intensities of NMR. An antiunitary mirror preserves every kernel and sends W to n − W. Product states: W counts the qubits with h_q + h′_q > 1. Spin-coherent reference: W counts the Majorana stars of the partner's symmetric part beyond a circle of latitude. The kernels are then the Fourier coefficients of the partner's Husimi function on the reference's circle of latitude. Haar-random partner: averaged over the partner, W(ψ, ·) equals the mean excitation number ⟨ψ|N|ψ⟩. For the reference |+⟩^⊗n, its variance is ζ(3/2)/(8√π)·√n to leading order. With one control qubit and copies of the two states, a certified protocol returns W exactly or abstains, using O(n log²n log(n/δ)/ν²) copy pairs. The margin ν is inverse-polynomial exactly for pairs that stay close under every common shift, and exponentially small for random partners. Classically, W is efficiently computable for product, matrix-product and computationally tractable states. Deciding W for circuit-prepared pairs at constant margin is BQP-complete. We claim no quantum advantage. 13 pages, 3 figures, 2 tables. The deposit includes the LaTeX source and a reference implementation in numpy. Its tests cover every theorem, proposition, lemma and corollary, and all figures and numbers are regenerated from a fixed seed.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
As small quantum computers are becoming available on different physical platforms, a benchmarking task known as cross-platform verification has been proposed that aims to estimate the fidelity of states prepared on two quantum computers. This task is fundamentally distributed, as no quantum communication can be performed between the two physical platforms due to hardware constraints, which prohibits a joint SWAP test. In this paper we settle the sample complexity of this task across all measurement and communication settings. The essence of the task, which we call distributed quantum inner product estimation, involves two players Alice and Bob who have k copies of unknown states ρ,σ (acting on ℂd) respectively. Their goal is to estimate Tr(ρσ) up to additive error ε∈(0,1), using local quantum operations and classical communication. In the weakest setting where only non-adaptive single-copy measurements and simultaneous message passing are allowed, we show that k=O(max{1/ε2,√d/ε}) copies suffice. This achieves a savings compared to full tomography which takes Ω(d3) copies with single-copy measurements. Surprisingly, we also show that the sample complexity must be at least Ω(max{1/ε2,√d/ε}), even in the strongest setting where adaptive multi-copy measurements and arbitrary rounds of communication are allowed. This shows that the success achieved by shadow tomography, for sample-efficiently learning the properties of a single system, cannot be generalized to the distributed setting. Furthermore, the fact that the sample complexity remains the same with single and multi-copy measurements contrasts with single system quantum property testing, which often demonstrate exponential separations in sample complexity with single and multi-copy measurements.
Copying an unknown quantum state into many approximate copies is known to become equivalent to estimating it and re-preparing: the best measure-and-prepare copier approaches the best quantum copier as the number of copies grows. We determine how fast. For equatorial qubit states, one input copy and m output copies, judged by the global fidelity, we prove F_EB/F_coh = 1 − 1/√(2m) + O(1/m). For phase states on d levels we prove, with explicit constants, the law 1 − F_EB/F_coh = (d − 1)/√(2m) + O(1/m), confirmed by exact computations up to m = 400; for qubits with T input copies the gap is √2·σ_T/√m, proved with explicit constants for T = 2, 3 (the upper bound conditionally on the optimality of the flat seed, R0(T); the lower bound unconditionally). The measure-and-prepare copier's deficit is the ground-state energy of a harmonic oscillator, one quantum per unknown phase. The exact quantum fidelity has plateaus at m = kd and kd + 1 (m = kd + 1 is the economical point). Numerical results suggest a period-d modulation of the ratio at order 1/m; a proof of its next-order coefficient remains open. This complements the known universality of the cloning fidelity itself, which depends only on the number of free parameters: within the phase-state family, the leading gap coefficient counts the independent phase directions. The classical values rest on the flat-seed theorem of the companion Paper 5, and the several-copy quantum values on the reduced coherent dual of the companion Paper 5. Core lemmas of the qubit proof and its explicit upper-bound constants are machine-checked in Lean 4; the remaining steps are proved on paper and independently refereed.