标题： 从模块化流中进行关系型整体重构 显示英文标题

标题： Relational bulk reconstruction from modular flow

摘要： AdS/CFT 中的纠缠楔子重建范式指出，对于边界子区域 $\bar{A}$ 的纠缠楔子内的一个体量子比特上的算符，可以将其重构为定义在 $\bar{A}$ 上的 CFT 算符。这自然适合于量子纠错框架，其中包含体量子比特的 CFT 态构成了一个对抗边界子区域 $A$ 擦除的编码。本文构建并研究了全息术中的关系体重建框架：给定两个都对抗边界区域 $A$ 擦除保护的码子空间，目标是关联这两个空间之间的算符重建。为了实现这一点，我们假设这两个码子空间由一组一参数族的码平滑连接而成，所有这些码都对抗 $A$ 的擦除，并且这些码上的最大纠缠态都是满秩的。我们认为这种码子空间可以在全息术的“基于测量”的设置中自然构造。在这种设置下，我们利用模理论推导出固定码子空间算符的重建流方程，原则上可以通过积分该方程来关联整个流中的重建算符。我们观察到我们的关系体重建公式与 Connes 共循环流的无限时间极限之间存在显著相似性，并采取了一些步骤使其联系更加严谨。我们还通过我们称之为模反射算符的典型重建映射提供了重建公式的替代推导。 显示英文摘要

摘要： The entanglement wedge reconstruction paradigm in AdS/CFT states that for a bulk qudit within the entanglement wedge of a boundary subregion $\bar{A}$, operators acting on the bulk qudit can be reconstructed as CFT operators on $\bar{A}$. This naturally fits within the framework of quantum error correction, with the CFT states containing the bulk qudit forming a code protected against the erasure of the boundary subregion $A$. In this paper, we set up and study a framework for relational bulk reconstruction in holography: given two code subspaces both protected against erasure of the boundary region $A$, the goal is to relate the operator reconstructions between the two spaces. To accomplish this, we assume that the two code subspaces are smoothly connected by a one-parameter family of codes all protected against the erasure of $A$, and that the maximally-entangled states on these codes are all full-rank. We argue that such code subspaces can naturally be constructed in holography in a "measurement-based" setting. In this setting, we derive a flow equation for the operator reconstruction of a fixed code subspace operator using modular theory which can, in principle, be integrated to relate the reconstructed operators all along the flow. We observe a striking resemblance between our formulas for relational bulk reconstruction and the infinite-time limit of Connes cocycle flow, and take some steps towards making this connection more rigorous. We also provide alternative derivations of our reconstruction formulas in terms of a canonical reconstruction map we call the modular reflection operator.