The information theoretic interpretation of the length of a curve
The information theoretic interpretation of the length of a curve
复制标题
曲线长度的信息论解释
DOI：
10.1007/jhep06(2015)157
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发表时间：
2014
影响因子：
5.4
通讯作者：
Brian Swingle
中科院分区：
文献类型：
--
作者：
B. Czech;P. Hayden;Nima Lashkari;Brian Swingle
在具有AdS3渐近的全息对偶性的背景下，Ryu-Takayanagi公式指出子区域的纠缠熵由特定体测地线的长度给出。纠缠熵可以被操作为将子区域的状态从一方传输到另一方同时保持与参考方的所有相关性所需的纠缠成本。那么问题来了，其他体积曲线的长度是否可以解释为其他一些信息论任务的纠缠成本。基于最近的结果表明，更一般的大块曲线的长度是由微分熵计算的，我们引入了一个新的任务，称为约束状态合并，其中边界子区域的状态必须使用由大块曲线的几何形状决定的位置和规模限制的操作来传输。我们的主要结果是，在约束状态合并条件下，传输子区域状态的代价由微分熵给出，因此对应的块曲线的符号长度。当成本为负时，约束状态合并提取而不是消耗纠缠。这个演示有两个部分：首先，我们展示了一个代价是曲线长度的协议，其次，我们证明了这个协议是最优的，因为它使用了最少的纠缠量。为了完成证明，我们还证明了具有大中心电荷的1+1维共形场理论中间隔的单次光滑条件熵可以很好地近似于它们的冯诺依曼对立物。我们还重新讨论了局部一致熵密度算子中微分熵和最大熵之间的关系，证明了这两个量在共形场理论中存在很大的定量差异。最后，我们对扩展和课程进行了简短的讨论。
In the context of holographic duality with AdS3 asymptotics, the Ryu-Takayanagi formula states that the entanglement entropy of a subregion is given by the length of a certain bulk geodesic. The entanglement entropy can be operationalized as the entanglement cost necessary to transmit the state of the subregion from one party to another while preserving all correlations with a reference party. The question then arises as to whether the lengths of other bulk curves can be interpreted as entanglement costs for some other information theoretic tasks. Building on recent results showing that the length of more general bulk curves is computed by the differential entropy, we introduce a new task called constrained state merging, whereby the state of the boundary subregion must be transmitted using operations restricted in location and scale in a way determined by the geometry of the bulk curve. Our main result is that the cost to transmit the state of a subregion under the conditions of constrained state merging is given by the differential entropy and hence the signed length of the corresponding bulk curve. When the cost is negative, constrained state merging distills entanglement rather than consuming it. This demon-stration has two parts: first, we exhibit a protocol whose cost is the length of the curve and second, we prove that this protocol is optimal in that it uses the minimum amount of entanglement. In order to complete the proof, we additionally demonstrate that single-shot smooth conditional entropies for intervals in 1+1-dimensional conformal field theories with large central charge are well approximated by their von Neumann counterparts. We also revisit the relationship between the differential entropy and the maximum entropy among locally consistent entropy density operators, demonstrating large quantitative discrepancy between the two quantities in conformal field theories. We conclude with a brief discussion of extensions and lessons.
影响因子：
2.4
作者：
Dupuis, Frederic;Berta, Mario;Renner, Renato
通讯作者：
Renner, Renato