Holographic subregion complexity from kinematic space
- 23 Downloads
We consider the computation of volumes contained in a spatial slice of AdS3 in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in a spatial slice of AdS3 as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in light of the complexity = volume proposal, which posits that complexity of holographic quantum states is computed by bulk volumes. An extension of this idea proposes that the holographic subregion complexity of an interval, defined as the volume under its Ryu-Takayanagi surface, is a measure of the complexity of the corresponding reduced density matrix. If this is true, our results give an explicit relationship between entanglement and subregion complexity in CFT, at least in the vacuum. We further extend many of our results to conical defect and BTZ black hole geometries.
KeywordsAdS-CFT Correspondence Gauge-gravity correspondence Black Holes in String Theory
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- C.M. Papadimitriou, Computational complexity, Addison-Wesley, Reading Massachusetts U.S.A. (1994).Google Scholar
- M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213 [quant-ph/0502070].
- L.A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley Publishing Company, Reading Massachusetts U.S.A. (1976).Google Scholar