Journal of High Energy Physics

, 2018:72 | Cite as

Holographic complexity is nonlocal

  • Zicao FuEmail author
  • Alexander Maloney
  • Donald Marolf
  • Henry Maxfield
  • Zhencheng Wang
Open Access
Regular Article - Theoretical Physics


We study the “complexity equals volume” (CV) and “complexity equals action” (CA) conjectures by examining moments of of time symmetry for AdS3 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 = αcχ, 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.


AdS-CFT Correspondence Black Holes 


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This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  2. 2.Department of PhysicsMcGill UniversityMontealCanada
  3. 3.School of PhysicsNankai UniversityTianjinChina

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