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 [-n E(R) + o(n)] for some function E(R). The E(R) is positive below some threshold R', which implies R' is a lower bound on the quantum capacity. The result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution, or from master equations.Comment: 19 pages, 2 figures. Ver.2: Comparisons with the previously known bounds and examples were added. Except for very noisy channels, this work's bound is, in general, better than those previously known. Ver.3: Introduction shortened. Minor change

Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction

A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.Comment: 16 pages, 3 figures. Ver.4: Title changed. Ver.3: Due to a request of the AE of the journal, the present version has become a combination of (thoroughly revised) quant-ph/0610194 and the former quant-ph/0610195. Problem formulations of polynomial complexity are strictly followed. An erroneous instance of a lower bound on minimum distance was remove