Complete information orders of quantum channels: equivalence and undecidability

Abstract

About complete information orders of finite-dimensional quantum channels. With unrestricted finite quantum references, it claims equivalence of relative-entropy, Holevo, BKM, SLD Fisher-information and measured-chi-squared comparisons, with order reversal under genuine Stinespring complementation. It further claims coRE-completeness for strictly positive rational-complex POVMs, the absence of a computable sufficient SLD-reference cutoff, and inapproximability of a bounded additive entropy defect at every fixed positive additive accuracy. The companion manuscript is From complete channel order to capacity and one-way distillation; its capacity and distillation conclusions depend on the order results developed here.