The pitfalls of deciding whether a quantum channel is (conjugate) degradable and how to avoid them

TL;DR

This paper develops a clear criterion for determining when a quantum channel's degradability status is unambiguous, extending known results and impacting quantum capacity calculations for certain channels.

Contribution

It introduces a necessary and sufficient condition for unambiguous degradability assessment, extending the antidegradability region for specific quantum channels.

Findings

Extended the antidegradability region for qubit and qutrit transpose depolarizing channels

Reproduced known results for qubit depolarizing channels

Established single-letter quantum capacity formulas for certain asymmetric cloners

Abstract

To decide whether a quantum channel is degradable is relatively easy: one has to find at least one example of a degrading quantum channel. But in general, no conclusive criterion exists to show the opposite. Using elementary methods we derive a necessary and sufficient condition to decide under what circumstances the conclusion is unambiguous. The findings lead to an extension of the antidegradability region for qubit and qutrit transpose depolarizing channels. In the qubit case we reproduce the known results for the class of qubit depolarizing channels (due to their equivalence). One of the consequences is that the optimal qubit and qutrit asymmetric cloners possess a single-letter quantum capacity formula. We also investigate the ramifications of the criterion for the search of exclusively conjugate degradable channels.