Holographic complexity is nonlocal

TL;DR

This work tests the CV and CA proposals for holographic complexity in AdS$_3$ wormholes with multiple boundaries, revealing that the formation complexity scales with the Euler characteristic as $ΔC = α c χ$ with universal, temperature- and moduli-independent coefficients. The CV and CA results, $ΔC_V = -\frac{4π}{3} c χ$ and $ΔC_A = \frac{1}{6} c χ$, show the complexity is governed by topology rather than detailed interior geometry, implying the dual CFT gate sets must be nonlocal. Furthermore, the paper argues that locality cannot fully account for holographic complexity, since bi-local gates are needed to entangle spatially separated thermofield-double-like regions in a way that does not affect the total complexity. A genus-bound scenario is discussed to preserve positivity of CA complexity, suggesting phase transitions to lower-genus bulk saddles and an energy-genus relation that constrains allowable topologies at finite temperature. Overall, the results highlight a topology-driven, nonlocal character of holographic complexity and connect bulk geometric features to constraints on dual quantum circuits.

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 $ΔC = αc χ$, where $c$ is the central charge and $χ$ is the Euler character of the bulk time-symmetric surface. The coefficients $α_V = -4π/3$, $α_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 $α_V$, $α_A$ 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 $α_A$, which requires the associated complexity to decrease when handles are added to our wormhole.

Figures (4)

• Figure 1: Left: The non-compact equilateral hyperbolic triangle geometry typical of the "islands" behind the horizon (solid lines) as a subset of the Poincaré disk. Center: Rough sketch of two such islands connected by a long tendril. Right: Rough sketch of an island with two of its own tendrils connected to each other.
• Figure 2: In this region, the causal shadow region (shaded) behind the horizon is exponentially thin. The spacetime approximates that of the two-sided BTZ black hole, and the dual CFT state is exponentially close to the thermofield double. Solid lines are the asymptotically (locally) AdS boundaries.
• Figure 3: The familiar conformal diagram for a 2-sided BTZ wormhole and its $t=0$ WdW patch (shaded). The dashed vertical line indicates the plane invariant under a right/left reflection symmetry. The geometry of any $W_i$ is identical to that of this WdW patch to (say) the left of the dashed line.
• Figure 4: A $t=0$ surface with $p+1$ boundaries (one solid line and $p$ dashed lines).