Holographic Complexity

TL;DR

Addresses holographic complexity for a subsystem in AdS/CFT by defining it as the volume bounded by the entanglement surface and computing it for a spherical region. Uses the RT prescription to derive the volume and its divergences, showing a leading volume-law term, a universal finite piece, and, for even dimensions, logarithmic contributions; examines excited states via AdS black holes with a quadratic correction in the deformation parameter. Finds that in the large-volume limit the complexity reduces to fidelity susceptibility for marginal perturbations, linking holographic complexity to quantum information measures and suggesting a central-charge-like universal term. Proposes a Wald-entropy-inspired generalization for higher-derivative gravities and outlines future directions including thermofield double setups and black-hole physics.

Abstract

For a field theory with a gravitational dual, following Susskind's proposal we define holographic complexity for a subsystem. The holographic complexity is proportional to the volume of a co-dimension one time slice in the bulk geometry enclosed by the extremal co-dimension two hyper-surface appearing in the computation of the holographic entanglement entropy. The proportionally constant, up to a numerical order of one factor is G R where G is the Newton constant and R is the curvature of the space time. We study this quantity in certain holographic model. We also explore a possible relation between the defined quantity and fidelity appearing in quantum information literature.