AI 中文总结
本研究完善近似量子纠错的通用恢复理论，证明单个恢复映射可校正给定误差集控制的所有通道，明确相关参数控制通用译码，并给出 Petz 映射作为显式通用恢复的性能保证。
AI 中文摘要
通用恢复指存在单个恢复映射可校正整个误差通道族，是量子纠错（QEC）的核心特征。在精确量子纠错中，线性性保证校正给定误差集的码也能校正所有 Kraus 算子属于其线性张成的通道，且单个恢复映射足以处理所有此类通道。传统上，将完美恢复放宽为带可控误差的近似量子纠错（AQEC）缺乏该结构。在近期论文（arXiv:2607.22995）中，我们建立的近似量子纠错理论表明，近似场景下存在受限形式的线性性，为给定误差集控制的通道族提供了统一 AQEC 保证。本研究完善了近似场景下通用恢复的另一半结论：单个恢复映射可同时校正给定误差集控制的所有通道。我们提出的误差集理论通过两个参数量化近似可校正性：控制最坏情况性能的环境泄漏距离，以及控制平均情况性能的 Knill–Laflamme 赫尔利距离。本文证明这两个量也控制通用译码，进一步研究与误差集自然关联的 Petz 映射作为显式通用恢复，在整个通道族上获得了统一的平均和最坏情况保证。
英文摘要
Universal recovery -- the existence of a single recovery map that corrects an entire family of error channels -- is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set. In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill--Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.
Commentsv2: 40pp, 4 figures. Added experimental data to support the theory
