Binding Complexity and Multiparty Entanglement

TL;DR

This work defines binding complexity as the minimal number of gates acting across multiple parties required to prepare a multipartite quantum state, and develops a comprehensive framework to compute it in a toy scalar-field model. Using Nielsen geometry with a cross-party gate cost, Euler-Arnold equations, and Gaussian-state analysis, the authors derive explicit results for Gaussian states and connect these complexity measures to holographic ideas about multiboundary wormholes and their interiors. They show that binding complexity scales linearly with entanglement entropy in several models, and propose a deep link to the interior volume of wormholes, refined by Euclidean path-integral constructions and bipartite graph structures. The paper also explores perturbative coherent-state deformations and double-trace interactions, illustrating how local couplings across parties generate binding complexity and potentially correspond to wormhole growth in holography. Overall, binding complexity emerges as a fine-grained probe of multiparty entanglement with intriguing holographic interpretations and concrete calculational strategies.

Abstract

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.

Figures (12)

• Figure 1: Quantum circuit diagram preparing the GHZ state from the factorized state $|0\rangle^{\otimes n}$. The box labeled $H$ indicates the Hadamard operator, a particular unitary one-qubit gate, while the symbol connecting lines refers to the CNOT operator, a unitary two-qubit gate. Here $\mathrm{CNOT} = |0\rangle \langle 0|_A \otimes 1_B + |1\rangle \langle 1|_A \otimes \sigma^x_B$ and $H = |+\rangle \langle 0| + |-\rangle \langle 1|$.
• Figure 2: We can "cut" a two-party gate (denoted by the red dashed line) by using its operator Schmidt decomposition into a sum of products of one-party operators.
• Figure 3: A sample piece of a unitary quantum circuit. The red lines denote the subsystem A and the blue lines denote the subsystem B. We have "cut" all the two-party gates acting across the bipartition by using their operator Schmidt decomposition into sums of products of one-party operators.
• Figure 4: When the entanglement entropy is large, the binding complexity varies linearly with the entropy up to exponentially small corrections. Here the number of parties is $m=12$. Other values of $m$ give similar results.
• Figure 5: The binding complexity varies linearly with the logarithm of the Gaussian negativity up to exponential corrections. Here the number of parties is again $m=12$ and other values of m give similar results.