Sector-Overlap Scores for Covariant Channels

A completely positive channel N that is covariant for a group G acting on a carrier W = ⊕_s V_s (pairwise inequivalent irreducibles) admits two natural bilinear read-outs of its sector structure: a "realign reading" A_phys[s,t] = Tr[P_s N(P_t)]/d_s, the channel's coefficient matrix on the sector algebra, and a "no-realign reading" A_ext[s,t] = (1/d_s) sum_{a,A} , a score. We prove three things. First, covariance forces N(P_t) to be block-scalar, so A_phys is the coefficient matrix c_{st} in N(P_t) = sum_s c_{st} P_s (Schur's lemma). Second, the score is the Gram matrix of the channel's sector-overlap vectors, A_ext[s,t] = (1/d_s) with w_s = (Tr[P_s K_k])_k; hence its diagonal is a manifest sum of squares (nonnegative for any channel) and D·beta is Hermitian positive semidefinite in the sector-trace metric D = diag(d_s), where beta = A_ext. Third, the score equals the sector-return diagonal, A_ext = diag(A_phys), if and only if two independent conditions hold: (I) the overlap vectors of distinct sectors are orthogonal, and (II) within each sector a trace/Frobenius balance sum_k |Tr K_k^{(ss)}|^2 = sum_k ||K_k^{(ss)}||_F^2. Condition (II) has a Schumacher entanglement-fidelity reading: the maximally-mixed entanglement fidelity of the sector-s diagonal sub-channel equals A_phys[s,s]/d_s. We call channels meeting (I)+(II) sector-return. A specific Sp(3) = USp(6)-covariant channel on the 35-dimensional sl(6) adjoint (split 14 + 21) is sector-return exactly, while generic covariant channels — the identity among them — are not; in the correct metric its diagonal sub-channels are scalar-unital and sit exactly at the (II)-balance, and a numerical spectral pass from exact-rational operators resolves them as Hermitian positive semidefinite of Choi rank D^2 = 36 (D = 6 the memory dimension), with dimension-indexed spectra whose leading eigenvalue is the sector-return coefficient. We close with the open problem this sharpens: whether covariant compression through a finite memory (a D-dimensional auxiliary memory, not a minimal Stinespring environment) forces sector-return.