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 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 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. 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 density operators, demonstrating large quantitative discrepancy between the two quantities in conformal field theories.
Highlights
Been the focus of a massive body of research1 driven by diverse theoretical and phenomenological motivations
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 the signed length of the corresponding bulk curve
Summary
We focus on pure three-dimensional anti-de Sitter space (AdS3). Our results apply in other asymptotically AdS3 geometries and in higher-dimensional holographic spacetimes, but they are subject to a number of technical caveats. We assume that this geometry arises as the dual description of the vacuum state of a conformal field theory (CFT) living on its asymptotic boundary — that is on an infinite line cross time. The Ryu-Takayanagi proposal [15, 16] relates the entanglement entropy of an interval I = (−a/2, a/2) in the CFT to the length of the spacelike geodesic, which asymptotes to the endpoints of I:. (A mathematically precise definition of constrained merging can be found in section 3.) Among all such procedures, the minimal number of Bell pairs is asymptotically given by (2.4), the length of the curve in Planck units Subject only to the prescribed constraints. (A mathematically precise definition of constrained merging can be found in section 3.) Among all such procedures, the minimal number of Bell pairs is asymptotically given by (2.4), the length of the curve in Planck units
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