Subsystem Complexity and Holography

TL;DR

This paper investigates holographic circuit complexity for spatial regions, comparing holographic proposals with tensor network models, and finds a qualitative match with the holographic action but not with volume for mixed states.

Contribution

It introduces and compares several holographic analogues of circuit complexity for mixed states, highlighting a promising match with the holographic action conjecture.

Findings

Holographic action complexity matches purification complexity for mixed states.

Holographic volume does not match any of the proposed mixed-state complexities.

Tensor network intuition supports the holographic action complexity results.

Abstract

We study circuit complexity for spatial regions in holographic field theories. We study analogues based on the entanglement wedge of the bulk quantities appearing in the "complexity = volume" and "complexity = action" conjectures. We calculate these quantities for one exterior region of an eternal static neutral or charged black hole in general dimensions, dual to a thermal state on one boundary with or without chemical potential respectively, as well as for a shock wave geometry. We then define several analogues of circuit complexity for mixed states, and use tensor networks to gain intuition about them. We find a promising qualitative match between the holographic action and what we call the purification complexity, the minimum number of gates required to prepare an arbitrary purification of the given mixed state. On the other hand, the holographic volume does not appear to match any of our definitions of mixed-state complexity.