Holographic complexity is nonlocal

TL;DR

This paper investigates holographic complexity in AdS_3 wormholes, revealing that complexity measures are nonlocal and independent of temperature and moduli, implying nonlocality in the dual CFT gate set.

Contribution

It demonstrates that holographic complexity coefficients are universal and independent of geometric parameters, indicating that the dual CFT complexity cannot be purely local.

Findings

Complexity coefficients are independent of temperature and moduli.

Complexity does not depend on separation of entangled patches.

Adding handles to wormholes decreases complexity.

Abstract

We study the "complexity equals volume" (CV) and "complexity equals action" (CA) conjectures by examining moments of of time symmetry for $\rm AdS_3$ wormholes having $n$ asymptotic regions and arbitrary (orientable) internal topology. For either prescription, the complexity relative to $n$ copies of the $M=0$ BTZ black hole takes the form $\Delta C = \alpha c \chi $, where $c$ is the central charge and $\chi$ is the Euler character of the bulk time-symmetric surface. The coefficients $\alpha_V = -4\pi/3$, $\alpha_A = 1/6 $ defined by CV and CA are independent of both temperature and any moduli controlling the geometry inside the black hole. Comparing with the known structure of dual CFT states in the hot wormhole limit, the temperature and moduli independence of $\alpha_V$, $\alpha_A$ implies that any CFT gate set defining either complexity cannot be local. In particular, the complexity of an efficient quantum circuit building local thermofield-double-like entanglement of thermal-sized patches does not depend on the separation of the patches so entangled. We also comment on implications of the (positive) sign found for $\alpha_A$, which requires the associated complexity to decrease when handles are added to our wormhole.