标题： 从模流中关系性批量重建 显示英文标题

标题： Relational bulk reconstruction from modular flow

摘要： 纠缠楔形重建范式在AdS/CFT中指出，对于边界子区域$\bar{A}$内的体量子比特，作用在体量子比特上的算符可以作为CFT算符在$\bar{A}$上重建。 这自然符合量子纠错的框架，其中包含体量子比特的CFT状态形成一个对抗边界子区域$A$擦除的编码。 在本文中，我们建立并研究全息论中的关系体重建框架：给定两个都对抗边界区域$A$擦除的编码子空间，目标是关联这两个空间之间的算符重建。 为了实现这一点，我们假设这两个编码子空间通过一个一参数族的编码平滑连接，所有这些编码都对抗$A$的擦除，并且这些编码上的最大纠缠态都是满秩的。 我们认为这样的编码子空间可以在全息论中以“基于测量”的设置下自然构造。 在此设置中，我们使用模理论推导出固定编码子空间算符的算符重建流动方程，原则上可以对其进行积分以在整个流动过程中关联重建的算符。 我们观察到我们的关系体重建公式与康尼斯余子流的无限时间极限之间有显著的相似之处，并采取了一些步骤来使这种联系更加严格。 我们还提供了我们重建公式的替代推导，这些推导是基于我们称之为模反射算符的标准重建映射。 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.