From complete channel order to capacity and one-way distillation

Abstract

A universal construction that converts failure of complete quantum-channel information order into strictly positive ordinary private and quantum capacity. It claims an exact-zero capacity undecidability classification for finite rational-complex memoryless channels and corresponding positivity and zero classifications for one-way distillable entanglement and secret key. The state reductions retain any prescribed two-way distillation lower bound. The construction explicitly charges reference transmission and activation helper uses to the constructed channel. Strict positivity is distinguished from any uniform positive capacity gap; fixed-error capacity inapproximability is not claimed. The exact order-equivalence and undecidability premises are developed in the companion manuscript Complete information orders of quantum channels: equivalence and undecidability.