Strong Converse for the Quantum Capacity of the Erasure Channel for Almost All Codes

A strong converse theorem for channel capacity establishes that the error probability in any communication scheme for a given channel necessarily tends to one if the rate of communication exceeds the channel’s capacity. Establishing such a theorem for the quantum capacity of degradable channels has been an elusive task, with the strongest progress so far being a so-called “pretty strong converse.” In this work, Morgan and Winter proved that the quantum error of any quantum communication scheme for a given degradable channel converges to a value larger than 1/ √ 2 in the limit of many channel uses if the quantum rate of communication exceeds the channel’s quantum capacity. The present paper establishes a theorem that is a counterpart to this “pretty strong converse.” We prove that the large fraction of codes having a rate exceeding the erasure channel’s quantum capacity have a quantum error tending to one in the limit of many channel uses. Thus, our work adds to the body of evidence that a fully strong converse theorem should hold for the quantum capacity of the erasure channel. As a side result, we prove that the classical capacity of the quantum erasure channel obeys the strong converse property. 1998 ACM Subject Classification H.1.1 Systems and Information Theory, E.4 Coding and Information Theory, Error control codes