Decoding quantum information via the Petz recovery map

TLDR: 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 is obtained.

Abstract: We obtain a lower bound on the maximum number of qubits, $Q^{n, \varepsilon}({\mathcal{N}})$, which can be transmitted over $n$ uses of a quantum channel $\mathcal{N}$, for a given non-zero error threshold $\varepsilon$. 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 {\em{Petz recovery map}} (also known as the {\em{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 $Q^{n, \varepsilon}({\mathcal{N}})$ scales as $\sqrt{n}$.