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arXiv:2607.12968

Maximal Classicalization of Finite-Group Quantum Reference-Frame Noise

Maxim V. Churilov

quant-phmath-phSubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations
v1 · Jul 14, 2026v2 · Aug 5, 2026
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Abstract

A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, $\inf_{Λ\in\mathrm{CPTP}} \frac{1}{2}\|Φ_q^U-Λ\circΦ_p^U\|_\diamond =\min_{r\in\mathcal{P}(G)}\frac{1}{2}\|q-r*p\|_1$. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance $\frac{1}{2}\|\operatorname{id}-Φ_u^U\|_\diamond=1-1/S(U)$ and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of $S_3$ yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.

Comments
10 pages, 1 figure. Ancillary verification code included

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A finite quantum reference token with group-valued misalignment induces a random-unitary channel, but optimal degradation is generally an optimization over all quantum post-processings. For a unitary representation U of a finite group G, we prove that the following conditions are equivalent: U contains every irreducible type; one ancilla-assisted input has an orthonormal G-orbit; signed group measures embed isometrically into channels in diamond norm; and, for every pair of noise laws p,q, inf⁡Λ∈CPTP12∥ΦqU−Λ∘ΦpU∥⋄=min⁡r∈P(G)12∥q−r∗p∥1\inf_{Λ\in\mathrm{CPTP}} \frac{1}{2}\|Φ_q^U-Λ\circΦ_p^U\|_\diamond =\min_{r\in\mathcal{P}(G)}\frac{1}{2}\|q-r*p\|_1infΛ∈CPTP​21​∥ΦqU​−Λ∘ΦpU​∥⋄​=minr∈P(G)​21​∥q−r∗p∥1​. Thus representation completeness is the exact carrier condition for universal reduction of quantum post-processing to classical convolution. We determine the minimum ancilla dimensions for an orthogonal orbit and for an invariant calibration seed. For an incomplete carrier, with visible Plancherel dimension S(U), we derive the exact conditional-expectation distance 12∥id⁡−ΦuU∥⋄=1−1/S(U)\frac{1}{2}\|\operatorname{id}-Φ_u^U\|_\diamond=1-1/S(U)21​∥id−ΦuU​∥⋄​=1−1/S(U) and an explicit quantum--classical deficiency gap. For irreducible carriers the deficiency is obtained in closed form; the faithful two-dimensional representation of S3S_3S3​ yields an exact ten-percent reduction relative to classical convolution. We also characterize law identifiability through the conjugation representation, provide finite linear programs and decision witnesses, and establish both a finite-dimensional obstruction and stable visible-band reconstruction for infinite compact groups. Deterministic ancillary code reproduces the finite-group examples and numerical regression checks.
Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.
A finite group-valued temporal reference frame defines a correlated random-unitary process rather than a list of independent channels. We identify a regime in which its complete trajectory law is an exact operational coordinate. If the carrier contains every irreducible representation of the finite group GGG, one ancilla-assisted input resolves all group elements, and for arbitrary laws μ,νμ,νμ,ν on GnG^nGn the strategy half-distance equals dTV(μ,ν)d_{\mathrm{TV}}(μ,ν)dTV​(μ,ν). Optimal output-side causal post-processing likewise reduces to classical convolution on GnG^nGn. This converts temporal hiding into a qqq-ary coding problem. Uniform coset laws of an [n,k]q[n,k]_q[n,k]q​ code are identical on every set of fewer than d(C⊥)d(C^\perp)d(C⊥) slots and perfectly distinguishable globally. For MMM sectors with leakage at most ηηη from ttt selected slots and global decoding error εεε, we obtain (1−ε)log⁡M≤(n−t)log⁡q+η+h2(ε)(1-ε)\log M \leq (n-t)\log q+η+h_2(ε)(1−ε)logM≤(n−t)logq+η+h2​(ε). For nested codes C⊂DC\subset DC⊂D, the payload R=dim⁡D−dim⁡CR=\dim D-\dim CR=dimD−dimC, invisible depth t=drel(C⊥,D⊥)−1t=d_{\mathrm{rel}}(C^\perp,D^\perp)-1t=drel​(C⊥,D⊥)−1, and sector distance drel(D,C)d_{\mathrm{rel}}(D,C)drel​(D,C) obey R+t+drel(D,C)−1≤nR+t+d_{\mathrm{rel}}(D,C)-1\leq nR+t+drel​(D,C)−1≤n and R+t+2e+f≤nR+t+2e+f\leq nR+t+2e+f≤n, where eee and fff are adversarial errors and known erasures. Nested generalized Reed-Solomon codes attain the bounds in their existence range. We also give the exact projection-rank leakage profile and show that hiding arbitrary coherent superpositions from ttt slots is precisely quantum erasure correction, yielding log⁡qK+2t≤n\log_q K+2t\leq nlogq​K+2t≤n. The results separate classical trajectory privacy from coherent temporal privacy and quantify the robust payload hidden from reduced process tomography.
Comparing two noisy quantum reference frames as statistical experiments depends on the dimension of the ancillary memory available to the decision procedure. For finite-dimensional channels A and B with invertible A, we show that exact simulation of all measurements assisted by an r-dimensional ancilla is equivalent to r-positivity of the unique factor Gamma = BA^{-1}. The hierarchy can be realized by physical channel pairs: every unital, trace-preserving map that is k-positive but not (k+1)-positive embeds as the factor between the channels D_a and Gamma composed with D_a on an exact interval determined by the smallest Choi eigenvalue. For depolarizing source and target channels D_a and D_b, including negative and singular source parameters, the phase boundary is Da⪰rDb⟺−1/(dr−1)≤b/a≤1\mathcal{D}_a \succeq_r \mathcal{D}_b \Longleftrightarrow -1/(dr-1) \leq b/a \leq 1Da​⪰r​Db​⟺−1/(dr−1)≤b/a≤1 for a≠0a\neq 0a=0. We derive closed formulas for the restricted level-r deficiency and for the distance to every physical post-processing, δphys(Db∣Da)=(1−1/d2)dist⁡(b,Ia)δ_{\mathrm{phys}}(\mathcal{D}_b\mid\mathcal{D}_a)=(1-1/d^2)\operatorname{dist}(b,I_a)δphys​(Db​∣Da​)=(1−1/d2)dist(b,Ia​), where Ia=conv⁡{a,−a/(d2−1)}I_a=\operatorname{conv}\{a,-a/(d^2-1)\}Ia​=conv{a,−a/(d2−1)}. The largest physical conversion cost hidden from all tests through level k is (d−k)/[d(d2−1)](d-k)/[d(d^2-1)](d−k)/[d(d2−1)]. An untouched m-level spectator changes the first detecting external level from k+1 to ⌊k/m⌋+1\lfloor k/m\rfloor+1⌊k/m⌋+1. A transpose--depolarizing construction shows that the separation is not confined to depolarizing factors. The results quantify the distinction between ancilla-restricted statistical simulation and implementation by a single quantum channel.
Abstract Random-unitary noise is an implementable way to hide quantum information, since it can be realized by applying unitary gates sampled from a classical random key. However, the unconstrained minimization of pairwise quantum α\alphaα-zzz R\'enyi divergences over all mixed-unitary channels is degenerate: the completely depolarizing channel is a Weyl average and maps every input state to Id/dI_d/dId​/d. This paper develops a non-erasing privacy--utility framework based on Pauli--Weyl randomization. For a finite protected ensemble E={ρ1,…,ρm}\mathcal{E}=\{\rho_1,\ldots,\rho_m\}E={ρ1​,…,ρm​}, a compact admissible set KKK of Weyl probability distributions, and a fidelity-type utility threshold η\etaη, we study the robust radius Rα,zη(E;K)=inf⁡p∈K: UE(Φp)≥ηmax⁡i≠jDα,z(Φp(ρi)∥Φp(ρj)).R^\eta_{\alpha,z}(\mathcal{E};K) = \inf_{p\in K:\,\mathcal{U}_{\mathcal{E}}(\Phi_p)\ge\eta} \max_{i\ne j} D_{\alpha,z}\bigl(\Phi_p(\rho_i)\Vert\Phi_p(\rho_j)\bigr). Rα,zη​(E;K)=infp∈K:UE​(Φp​)≥η​maxi=j​Dα,z​(Φp​(ρi​)∥Φp​(ρj​)). We prove well-posedness, a zero-radius characterization, and active-pair optimality conditions. The main exact result is a Pauli reduction theorem for equal-purity binary qubit ensembles: in adapted Bloch coordinates, optimization over all Pauli channels reduces to a two-parameter contraction problem separating the sensitive direction from the common direction. This yields an explicit two-eigenvalue formula for the reduced α\alphaα-zzz divergence and a closed-form optimal mechanism on the fidelity line α=z=1/2\alpha=z=1/2α=z=1/2 for arbitrary pure binary qubit states. A complementary commuting result solves antipodal binary pairs throughout the data-processing region. For diagonal qudit ensembles, Weyl randomization reduces to cyclic convolution; the utility has a Fourier autocorrelation representation, and the minimax problem admits convex global formulations for 0<α<10<\alpha<10<α<1 and α=2\alpha=2α=2. Finally, the radius is shown to be stable under post-processing and measurement, to imply binary discrimination bounds, and to support hardware-aware calibration certificates. MSC Classification: Primary 81P45 , 94A17; Secondary 81P50 , 47N50 , 49K35 , 90C25 ,90C30
Understanding the limits of quantum communication is a central problem in quantum information theory. Structural properties of quantum channels provide tools for analyzing these limits. In particular, approximate degradability links quantum-capacity bounds to the accuracy with which a channel's environment can be reconstructed from its output. We study this reconstruction problem for noisy Landau-Streater channels across finite-spin representations. For spin jjj and noise probability ppp, we construct an explicit degrading map with diamond-norm error at most [2+8/(j(j+1))]p2≤(38/3)p2[2+8/(j(j+1))]p^2\le(38/3)p^2[2+8/(j(j+1))]p2≤(38/3)p2 on the common interval 0≤p≤1/80\le p\le1/80≤p≤1/8. A lower bound establishes quadratic order for this construction at each fixed spin. Floating-point semidefinite optimization places its error within 2.48%2.48\%2.48% of the numerical optimum on the sampled grid for four spins. Covariance and conditional-entropy concavity yield explicit two-sided quantum-capacity bounds for this family with a spin-uniform remainder. Building on the low-noise Pauli cancellation strategy, we formulate a sufficient condition based on a common scalar adjoint action on fixed, possibly nonunitary noise Kraus operators. The resulting reconstruction-error bound is uniform in system dimension and Kraus count. The criterion applies to irreducible SU(N)SU(N)SU(N) generator channels and noisy Werner-Holevo channels, with the SU(3)SU(3)SU(3) adjoint representation providing an example outside the mixed-unitary class.
This survey introduces the role of frame phase retrieval in pure-state identification and information preservation by quantum channels. For unit vectors, the lift x↦xx∗x\mapsto xx^*x↦xx∗ identifies vectors that differ only by a global phase with the same rank-one quantum state, converts frame intensities into linear functionals of the lifted state, and makes the adjoint pullback of output observables an operator-valued measurement on the input system. From this viewpoint, a channel is phase retrievable exactly when it is injective on pure states. We develop this correspondence through Choi-rank-two linear combinations of Kraus operators, higher-rank relative spectra, structural obstructions, and constructions with prescribed Choi rank. We distinguish injective identification from full tomography, perfect one-shot discrimination, zero-error classical communication, and exact quantum correction. Twirling channels then provide a structured setting in which commutants, irreducible dimensions, multiplicities, orbit-frame orthogonality, coding indices, and phase-retrievable subspaces can be read from group representations.