General Bounds on Holographic Complexity

TL;DR

This paper proves a positive volume theorem for asymptotically AdS spacetimes, establishing a lower bound on holographic complexity and providing insights into the geometric and energetic properties of such spacetimes.

Contribution

It introduces a positive volume theorem for asymptotically AdS spacetimes and extends the positive energy theorem concepts to holographic complexity, with proofs in four dimensions and evidence for higher dimensions.

Findings

Maximal volume slices have nonnegative vacuum-subtracted volume.

Wormholes with a fixed throat are more complex than AdS-Schwarzschild.

The results support the Complexity=Volume conjecture and relate to Lloyd's bound.

Abstract

We prove a positive volume theorem for asymptotically AdS spacetimes: the maximal volume slice has nonnegative vacuum-subtracted volume, and the vacuum-subtracted volume vanishes if and only if the spacetime is identically pure AdS. Under the Complexity=Volume proposal, this constitutes a positive holographic complexity theorem. The result features a number of parallels with the positive energy theorem, including the assumption of an energy condition that excludes false vacuum decay (the AdS weak energy condition). Our proof is rigorously established in broad generality in four bulk dimensions, and we provide strong evidence in favor of a generalization to arbitrary dimensions. Our techniques also yield a holographic proof of Lloyd's bound for a class of bulk spacetimes. We further establish a partial rigidity result for wormholes: wormholes with a given throat size are more complex than AdS-Schwarzschild with the same throat size.