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The Shift Winding: An Integer Invariant of a Pair of Quantum States that Phase-Insensitive Overlap Kernels Cannot See

Özcan Kasal

Published Oct 15, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23384430

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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.
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.
A theoretical and numerical study of when the anti-holomorphic (conjugate-variable) dependence of a two-point transfer kernel is forced by the underlying open-system dynamics rather than removable by gauge, basis, or preparation. For two counter-winding mode ladders coupled unidirectionally (reservoir-engineered, Lindblad-type, per-pair Jordan double-pole structure), the transfer kernel sums to a closed form involving the Lerch transcendent in the product of conjugated coordinates. Reciprocity is shown to be equivalent to the existence of an exchange-Hermitian mirror partner of the kernel; unidirectionality removes that partner. All symbolic claims are certified by exact Wirtinger-derivative computation. A recirculating-fiber-loop implementation and a time-domain protocol are proposed, with noise/imperfection analysis and explicit falsifiability criteria. Version 3, additive over v2 — the boundary arc (Sections 3–5): the kernel's intrinsic boundary at |x| = 1 with its generic-existence statement (the wall belongs to the class; its location is set by the tail of the weights); the chiral phase law of the branch-cut jump (machine-exact ratio, linear drift at rate kappa/2delta, family-independent); the log-periodic interference calendar on the log-scale coordinate, with prior-art positioning against discrete scale invariance (Sornette; Derrida–Itzykson–Luck); the Fourier floor theorem with full proof (the residual relief of large kernel sums equals the window-contrast of the Fourier transform of the weighted winding measure; finite atomic ladders keep the calendar forever by Bohr almost-periodicity, continua forget by Riemann–Lebesgue); closed-form floors (Ci/Si integrals, cardinal sine); and an inverse spectroscopy reading the hidden weighted distribution of winding speeds from the measured relief, with proven limits (exact mirror blindness of modulus-only data; window = aperture). Non-Hermitian point-gap topology references added (Wanjura et al. 2020; Zirnstein et al. 2021). Title-date correction (July 10, 2026). No physical claim is added; all results are machine-verified structural characterizations. Note added at deposit time: subsequent to the sealing of this version, the mirror-blindness limit of the inverse spectroscopy has been lifted by a reference-needle heterodyne protocol (demodulation law, asymmetry law, quadrature recovery of the full complex transform), machine-verified and judge-certified; traces and harnesses are in the public repository below. This upgrade will be the subject of the next linked deposit. All FINDINGS and machine harnesses backing every number in Sections 3–5, plus an independent external from-scratch reproduction audit, are public: https://github.com/antparis/oneway-kernel (MIT).