Holographic Subregion Complexity from Kinematic Space

TL;DR

This paper derives a formula connecting bulk volumes in AdS3 to entanglement entropies in the dual CFT, providing insights into holographic subregion complexity and its relation to entanglement.

Contribution

It presents an explicit integral formula for bulk volumes in AdS3 using kinematic space and relates subregion complexity to entanglement entropies in the CFT.

Findings

Derived a volume formula in terms of kinematic space

Expressed subregion volume using entanglement entropies

Extended results to conical defect and BTZ geometries

Abstract

We consider the computation of volumes contained in a spatial slice of AdS$_3$ 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 the spatial slice 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.