Brian Swingle (Stanford) “The Information Theoretic Interpretation of the Length of a Curve” (String Seminars)

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Abstract: 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 other bulk curves have an interpretation as an entanglement cost in some information theoretic protocol. Building on recent results showing that the length of more general bulk curves is computed by the differential entropy (which is a linear combination of entanglement entropies associated with a partitioning of a boundary region into subregions), we introduce a new task called constrained state merging which is also associated with a partitioning of a boundary region into subregions. 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 length of the corresponding bulk curve. This demonstration 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. Finally, all costs above technically refer to the asymptotic cost per copy in the limit where many copies of the subregion state are transmitted, but this is an artificial limit. We complete the story by showing that approximately the same cost is necessary for transmitting only a single copy of the state, a generally much more difficult task. We conclude with a brief discussion of extensions and lessons.