Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower bounded by 1− exp[−nE(R) + o(n)] for some function E(R). The E(R) is positive below some threshold R0, a direct consequence of which is that R0 is a lower bound on the quantum capacity. This is an extension of the author’s previous result [M. Hamada, Phys. Rev. A, vol. 65, 052305, 2002; LANL e-Print, quant-ph/0109114, 2001]. While it states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.