Journal of High Energy Physics

, 2019:69 | Cite as

Binding complexity and multiparty entanglement

  • Vijay Balasubramanian
  • Matthew DeCross
  • Arjun KarEmail author
  • Onkar Parrikar
Open Access
Regular Article - Theoretical Physics


We introduce “binding complexity”, a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show that our states can be prepared by the Euclidean path integral in (0 + 1)-dimensional quantum mechanics on graphs with wormhole-like structure. We compute the binding complexity of our states by adapting the Euler-Arnold approach to Nielsen’s geometrization of gate counting, and find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also compute the binding complexity of general coherent states in perturbation theory, and show that for “double-trace deformations” of the Hamiltonian the effects resemble expansion of a wormhole interior in holographic theories.


AdS-CFT Correspondence Black Holes 


Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.David Rittenhouse LaboratoryUniversity of PennsylvaniaPhiladelphiaU.S.A.
  2. 2.Theoretische Natuurkunde, Vrije Universiteit Brussel (VUB), and International Solvay InstitutesBrusselsBelgium

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