Optimal recovery for quantum error correction

Abstract

The calculation of the error threshold of quantum error correcting codes typically proceeds as follows. First, syndromes are measured. Then, a decoder infers the error chain and the corresponding correction is applied. The threshold is then defined as the largest correctable error rate, with the maximum-likelihood decoder corresponding to the ``optimal'' threshold. However, a broader set of operations could be used to recover quantum information. The true optimal threshold should be optimised over all possible recovery schemes, which can be described by quantum channels. Here, we study such optimal recovery channels and their thresholds $p_\mathrm{th}^\mathrm{opt}$. We introduce an information-theoretic quantity, mutual trace distance, which provides a necessary and sufficient diagnostic for sharply determining $p_\mathrm{th}^\mathrm{opt}$ without explicit optimisation. In contrast, previous works give a lower bound on $p_\mathrm{th}^\mathrm{opt}$ by specifying particular recovery schemes, e.g. Schumacher-Westmoreland (SW) which provides coherent information as a diagnostic to lower bound $p^\mathrm{opt}_\mathrm{th}$. We prove that the Petz and SW recovery schemes are optimal, i.e. their threshold is $p_\mathrm{th}^\mathrm{opt}$. With their optimality established, we explore the structure of optimal and non-optimal recovery schemes and their phase diagrams.

Figures (2)

• Figure 1: Schematic of some main results. First, we show that the mutual trace distance $T'_{R:E}$ (\ref{['main:def:mutual-trace-distance']}) is a necessary and sufficient quantity to detect the optimal threshold over all recovery schemes, $p^\mathrm{opt}_\mathrm{th}$, whose performance is measured by the entanglement infidelity $1 - F_\mathrm{e}$. Next, we show that the Petz and Schumacher-Westmoreland recovery schemes $\mathcal{R}^\mathrm{Petz}$, $\mathcal{R}^\mathrm{SW}$, are optimal in that their threshold exactly coincides with the optimal one, $p^\mathrm{opt}_\mathrm{th} = p^\mathrm{Petz}_\mathrm{th} = p^\mathrm{SW}_\mathrm{th}$. $p$ is the noise strength.
• Figure 2: Illustration of \ref{['main:lem:non-optimal-recovery']} for a schematic recoverability phase diagram. Since Petz recovery maps with $p_* = p_\mathrm{s}$ is the optimal, any parameter on the horizontal of the recovery optimal line must recover as equally or worse.

Theorems & Definitions (22)

• Definition 1: Optimal recovery channel
• Theorem : Two-way bound of the Petz recovery channel, Theorem 2 of Ref. barnum2002reversing
• Definition 2: Recovery scheme and threshold
• Corollary 1: Optimality of the Petz recovery scheme
• Theorem : Lower bounds of entanglement fidelity of SW recovery channel, Ref. schumacher2002approximate
• Theorem 1: Upper bound for optimal entanglement fidelity
• Definition 3: Mutual trace distance
• Corollary 2: Two-way bounds for coherent information
• Theorem 2: Optimality of the Schumacher-Westmoreland recovery scheme
• proof