Finite-field Clifford symmetries and single-letter coherent information of stabilizer convolution channels

Abstract

Stabilizer convolution channels in odd-prime dimensions have a natural description in finite-field phase space. We analyze these channels using Clifford symmetry and show that, for a fixed environmental state, the maximal single-letter coherent information is unchanged under Clifford-equivalent choices of the stabilizer convolution matrix. This reduces the apparently quadratic family of admissible stabilizer convolutions, at the one-use level, to sign-orbit classes of a single finite-field parameter. We also isolate quadratic cases in which the induced Clifford symmetry becomes either the identity or phase-space inversion; these cases give symmetry-protected families with zero maximal single-letter coherent information. To complement this symmetry analysis, we introduce a minimal weighted magic entropy and prove that it sharpens the magic-relative-entropy upper bound for positive convolution channels at the single-letter level. Finally, we define the single-letter coherent-information power of a convolution and show that the optimizing environmental state may be taken to be pure. Small-dimensional numerical examples illustrate the sign-orbit classification and show that coherent-information enhancement is governed by the finite-field convolution parameter rather than by dimension alone.