论文标题
离散的批量重建
Discrete Bulk Reconstruction
论文作者
论文摘要
根据广告/CFT的对应关系，某些空间的几何形状由生活在其边界上的量子状态完全确定，实际上，冯·诺伊曼（Von Neumann）部分的边界状态熵。这项工作研究了多项式时间从熵中重建几何形状的程度。 Bouland，Fefferman和Vazirani（2019）认为，如果一个人想重建黑洞的室内装饰区域，则广告/CFT地图可能成倍复杂。我们的主要结果提供了一种相反的结果：我们表明，在单个1D边界的特殊情况下，如果输入数据由连续边界区域的熵列表组成，并且如果熵满足了称为强亚热的单个不平等，那么我们可以在线性时间内构造整体图形模型。此外，批量图是平面，它具有$ o（n^2）$顶点（信息理论最小值），并且是``通用''，仅具有边缘权重，具体取决于所讨论的特定熵。从组合的角度来看，我们的问题归结为著名的最小切割问题的``反向''：而不是被给出图形并要求找到最小的切割，而是在这里允许我们分开各种顶点的刻板值，并且需要找到与这些值一致的加权图形的图形。我们解决此问题的解决方案依赖于``Bulkless''图的概念，该图可能对ADS/CFT具有独立的兴趣。我们还在多个1D边界的情况下取得了初步的进步 - 在其中可以通过虫洞连接边界 - 包括$ O（n^4）$ VERTICES的上限，每当存在平面批量图时（因此将问题放入复杂性类别$ \ m athssf {np} $）。
According to the AdS/CFT correspondence, the geometries of certain spacetimes are fully determined by quantum states that live on their boundaries -- indeed, by the von Neumann entropies of portions of those boundary states. This work investigates to what extent the geometries can be reconstructed from the entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued that the AdS/CFT map can be exponentially complex if one wants to reconstruct regions such as the interiors of black holes. Our main result provides a sort of converse: we show that, in the special case of a single 1D boundary, if the input data consists of a list of entropies of contiguous boundary regions, and if the entropies satisfy a single inequality called Strong Subadditivity, then we can construct a graph model for the bulk in linear time. Moreover, the bulk graph is planar, it has $O(N^2)$ vertices (the information-theoretic minimum), and it's ``universal,'' with only the edge weights depending on the specific entropies in question. From a combinatorial perspective, our problem boils down to an ``inverse'' of the famous min-cut problem: rather than being given a graph and asked to find a min-cut, here we're given the values of min-cuts separating various sets of vertices, and need to find a weighted undirected graph consistent with those values. Our solution to this problem relies on the notion of a ``bulkless'' graph, which might be of independent interest for AdS/CFT. We also make initial progress on the case of multiple 1D boundaries -- where the boundaries could be connected via wormholes -- including an upper bound of $O(N^4)$ vertices whenever a planar bulk graph exists (thus putting the problem into the complexity class $\mathsf{NP}$).
