[论文解读] Petz recovery from subsystems in conformal field theory

Shreya Vardhan, Annie Y. Wei+1 more|arXiv (Cornell University)|Jul 26, 2023

Quantum many-body systems被引用 4

一句话总结

本文研究了在1+1维共形场论（CFT）真空态中，利用twirled Petz映射（一种从更小子系统的约化密度矩阵重建密度矩阵的量子通道）恢复多体纠缠的性质。结果表明，原始态与恢复态之间的保真度、相对熵和迹距离度量具有普遍性——仅依赖于CFT的中心电荷和区间交比，而不依赖于算符内容，展示了紫外有限性以及重整化群重整化与OPE极限的非对易性。

ABSTRACT

We probe the multipartite entanglement structure of the vacuum state of a CFT in 1+1 dimensions, using recovery operations that attempt to reconstruct the density matrix in some region from its reduced density matrices on smaller subregions. We use an explicit recovery channel known as the twirled Petz map, and study distance measures such as the fidelity, relative entropy, and trace distance between the original state and the recovered state. One setup we study in detail involves three contiguous intervals $A$, $B$ and $C$ on a spatial slice, where we can view these quantities as measuring correlations between $A$ and $C$ that are not mediated by the region $B$ that lies between them. We show that each of the distance measures is both UV finite and independent of the operator content of the CFT, and hence depends only on the central charge and the cross-ratio of the intervals. We evaluate these universal quantities numerically using lattice simulations in critical spin chain models, and derive their analytic forms in the limit where $A$ and $C$ are close using the OPE expansion. In the case where $A$ and $C$ are far apart, we find a surprising non-commutativity of the replica trick with the OPE limit. For all values of the cross-ratio, the fidelity is strictly better than a general information-theoretic lower bound in terms of the conditional mutual information. We also compare the mutual information between various subsystems in the original and recovered states, which leads to a more qualitative understanding of the differences between them. Further, we introduce generalizations of the recovery operation to more than three adjacent intervals, for which the fidelity is again universal with respect to the operator content.

研究动机与目标

• 探究1+1维CFT真空态的多体纠缠结构，超越双体纠缠熵的范畴。
• 研究恢复操作是否能够重建不通过中间子系统介导的非相邻区域之间的关联。
• 确定原始态与恢复态之间距离度量是否具有普遍性，即是否独立于CFT的算符内容。
• 评估twirled Petz映射在恢复三个或更多连续区间的态时的性能。

提出的方法

• 使用twirled Petz映射作为显式恢复通道，从更小子区域的约化密度矩阵重建区域的密度矩阵。
• 采用保真度、相对熵和迹距离作为度量，量化恢复质量。
• 分析三个连续区间A、B、C，以分离不通过B介导的A与C之间的关联。
• 在临界自旋链中使用格点模拟，数值计算普遍恢复度量。
• 在A与C接近的OPE极限下，利用算符乘积展开技术推导解析表达式。
• 使用重整化技巧和夹层Rényi相对熵分析渐近行为，并研究其与OPE极限的非对易性。

实验结果

研究问题

• RQ1 twirled Petz映射能否以普遍精度，从未在更小子系统上的约化密度矩阵中恢复1+1维CFT的真空态？
• RQ2 恢复保真度和距离度量是否独立于CFT的算符内容，仅依赖于中心电荷和区间几何结构？
• RQ3 在两个区间接近的极限（OPE极限）下，恢复度量的行为如何？其与重整化技巧的比较结果如何？
• RQ4 恢复保真度是否超过基于条件互信息的一般信息理论下界？
• RQ5 原始态与恢复态之间的互信息结构有何差异？这揭示了何种非马尔可夫关联？

主要发现

• 原始态与恢复态之间的保真度、相对熵和迹距离具有紫外有限性与普遍性——仅依赖于中心电荷和区间的交比，而不依赖于CFT的算符内容。
• 在A与C接近的OPE极限下，恢复保真度被解析推导，显示出由中心电荷决定的普遍标度行为。
• 在A与C广泛分离的情况下，发现重整化技巧与OPE极限之间存在出人意料的非对易性。
• 恢复保真度严格超过基于条件互信息的一般信息理论下界，表明其可恢复性强于预期。
• 将恢复通道推广至三个以上相邻区间的推广形式，其普遍性仍保持不变，不依赖于算符内容。
• 原始态与恢复态之间互信息的比较揭示了定性差异，凸显了恢复通道未能捕捉的非马尔可夫关联的存在。