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.
The Petz recovery map (or transpose channel) was introduced by Petz @cite_40 @cite_43 (see also @cite_31 ). @cite_46 it was shown that, if the Petz recovery map is used as the decoding operation, then the average error incurred in sending an ensemble of commuting states through a quantum channel is at most twice the minimum error. Later, this map was also used to characterize so-called quantum Markov chain states @cite_1 . Furthermore, it was used to study quantum error correcting codes in @cite_12 .
@cite_31 @cite_1 @cite_43 @cite_40 @cite_46 @cite_12
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We demonstrate that there exists a universal, near-optimal recovery map—the transpose channel—for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement. 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. We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information. We construct a practically implementable classical processing for the Bennett-Brassard 1984 (BB84) protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high a key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25 limit.