Decoding quantum information via the Petz recovery map

2015-04-17

arXiv.org

AI Summary

This work addresses the quantum capacity achievability problem for a quantum channel $mathcal{N}$ under finite blocklength $n$ and non-zero error threshold $varepsilon$, i.e., determining the maximum number $Q_{n,varepsilon}(mathcal{N})$ of qubits reliably transmissible using $n$ channel uses. We propose a novel decoding construction combining single-shot entanglement transmission bounds, second-order asymptotic expansion, and the Petz recovery map (transpose channel). For the first time, we rigorously prove that this map achieves the coherent information rate in asymptotic quantum error correction. This yields a second-order asymptotic achievability bound on $Q_{n,varepsilon}(mathcal{N})$, revealing its scaling behavior: sublinear $sqrt{n}$-dependence below the error threshold, and—surprisingly—linear scaling $Q_{n,varepsilon} sim n$ above it (e.g., for the 50–50 erasure channel). Our results provide a unified characterization of the precision–efficiency trade-off in quantum communication under finite resources.

Abstract

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.

Problem

• Quantum Channel Capacity
• Quantum Bits Transmission
• Error Tolerance

Innovation

• Petz recovery map
• quantum channel capacity
• second-order asymptotics