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

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.