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

Abstract

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 R 0 , a direct consequence of which is that R 0 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. KeywordsCompletely positive linear maps, error exponent, fidelity, symplectic geometry, the method of types, quantum capacity, quantum error-correcting codes.

Information Rates Achievable With Algebraic Codes on Quantum Discrete Memoryless Channels

“…We remark that this work's bound holds true for general discrete memoryless channels (TPCP maps) as treated in [11]. Namely, if we associate the probability distribution P = P A with a channel A [or P = P U A with some TPCP map U on L(H)] as in [11,Section II], then the bound in (32) and that in Corollary 1 are true for this channel.…”

“…The theorem is proved with a random coding argument similar to those in [10], [11], the main difference being in the decoding strategy. A concatenated code associated with cat(L, L out ) is a symplectic stabilizer code, so that we can apply the decoding strategy described in Section III-D to it.…”

“…A concatenated code associated with cat(L, L out ) is a symplectic stabilizer code, so that we can apply the decoding strategy described in Section III-D to it. Especially, minimum entropy decoder employed in [10], [11] can be used. In the proof of Theorem 2, however, we modify this decoding strategy incorporating Shor and Smolin's idea.…”

“…Namely, if we associate the probability distribution P = P A with a channel A [or P = P U A with some TPCP map U on L(H)] as in [11,Section II], then the bound in (32) and that in Corollary 1 are true for this channel. This can be shown in a quite similar way to that in [11].…”

“…On the other hand, known lower bounds appear to have left much room for improvement. For example, on the capacity of the depolarizing channel, which suffers uniform depolarization and can be specified by Kraus operators √ 1 − pI, p/3X, p/3Y, p/3Z with I and X, Y, Z being the identity and Pauli operators, respectively, the highest lower bound known is 1 − h(p) − p log 2 3 for a wide range of p, where h is the binary entropy function [7], [8], [9], [10], [11]. Shor and Smolin [12] argued this bound is not tight showing the existence of concatenated quantum codes that slightly go beyond it for a limited range of p, which revealed a remarkable feature of the issue of the quantum capacity.…”