A lower bound on the quantum capacity of channels with correlated errors

Abstract

The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1 − exp[−nE(R) + o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R 0 , which implies R 0 is a lower bound on the quantum capacity. This work is an extension of the author's previous works [M. Hamada, Phys. Rev. A, 65, 052305 (2002), e-Print quant-ph/0109114, LANL, 2001, and M. Hamada, e-Print quant-ph/0112103, LANL, 2001], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this paper to those in the previous works are similar to that of Markov chains to sequences of independent identically distributed random variables.

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

“…Extending this work's result to the case of channels with memory of a Markovian nature is possible if second-order (or higher-order) types are used instead of the usual types [52]. It may be also interesting to ask whether the present approach will help us obtain bounds or improve the known ones for Gaussian quantum channels already discussed in the literature [53], [54], [55].…”

Quantum channels with memory

“…[18,19] and references therein). A Lindbladian approach to memory channels has been taken by Daffer et al [20,21].…”

Witnessing quantum capacities of correlated channels

“…Correlated qubit channels were originally studied in terms of classical information transmission and it was shown that for certain ranges of the correlation strengths the use of entanglement allows one to enhance the amount of transmitted information along the channel [7]. Quantum memory (or correlated) channels then attracted growing attention, and interesting new features emerged by modeling of relevant physical examples, including depolarizing channels [8], Pauli channels [9][10][11], dephasing channels [12][13][14][15][16], amplitude damping channels [17,18], Gaussian channels [19], lossy bosonic channels [20,21], spin chains [22], collision models [23] and a micro-maser model [24] (for a recent review on quantum channels with memory effects see Ref. [25]).…”