Maximal Classicalization of Finite-Group Quantum Reference-Frame Noise

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, ${\inf }_{\Lambda \in \mathrm{CPTP}}\frac{1}{2}\Vert {\Phi }_q^U-\Lambda \circ {\Phi }_p^U{\Vert }_{\diamond }={\min }_{r\in \mathcal{P}(G)}\frac{1}{2}\Vert q-r\ast p{\Vert }_1$. 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 $\frac{1}{2}\Vert \mathrm{id}-{\Phi }_u^U{\Vert }_{\diamond }=1-1/S(U)$ 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.

13 pages, 1 figure. Ancillary verification code included

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