We obtain a lower bound on the maximum number of qubits, Qn, e(N), which can be transmitted over n uses of a quantum channel N, for a given non-zero error threshold e. To obtain our result, we first derive a bound on the one-shot entanglement transmission capacity of the channel, and then compute its asymptotic expansion up to the second order. In our method to prove this achievability bound, the decoding map, used by the receiver on the output of the channel, is chosen to be the Petz recovery map (also known as the transpose channel). Our result, in particular, shows that this choice of the decoder can be used to establish the coherent information as an achievable rate for quantum information transmission. Applying our achievability bound to the 50-50 erasure channel (which has zero quantum capacity), we find that there is a sharp error threshold above which Qn, e(N) scales as n.
Our work should be considered as a new step towards understanding the usefulness of the Petz recovery map. In particular it would be interesting to see whether the ideas in our work can be used to show tight achievability bounds for other quantum protocols such as quantum state merging and quantum state redistribution. Another open question in this area is the optimality of the Petz recovery map in the Fawzi-Renner inequality @cite_8 for approximate Markov chain states (see @cite_10 and references therein for a discussion of this question).
@cite_10 @cite_8
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The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R∘N)(ρ), where R is a recovery map with the property that σ=(R∘N)(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction. A state on a tripartite quantum system ( A B C ) forms a Markov chain if it can be reconstructed from its marginal on ( A B ) by a quantum operation from B to ( B C ). We show that the quantum conditional mutual information I(A : C|B) of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.