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arXiv:2411.03422

Signals of multiparty entanglement and holography

Vijay Balasubramanian, Monica Jinwoo Kang, Chitraang Murdia, Simon F. Ross

Published Nov 5, 2024• 0 citations
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Abstract

We study multiparty entanglement signals, which are functions of a quantum state that are non-zero only when the state has multiparty entanglement. We consider known signals of three- and four-party entanglement, and propose new signals for four- and higher-party entanglement. We make some remarks on their general properties, but mainly focus on using holographic states in AdS$_3/$CFT$_2$ as a test case to explore their properties. For both the AdS vacuum and multiboundary wormhole states, we find that the multiparty entanglement signals are generically non-zero and of order one in units of the central charge of the dual CFT, revealing substantial multiparty entanglement. In the large-horizon limit of multiboundary wormhole states, however, the signals for three or more parties become small indicating the short-range nature of entanglement.

DOI
10.48550/arxiv.2411.03422

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A bstract We study multiparty entanglement signals, which are functions of a quantum state that are non-zero only when the state has multiparty entanglement. We consider known signals of three- and four-party entanglement, and propose new signals for four- and higher-party entanglement. We make some remarks on their general properties, but mainly focus on using holographic states in AdS 3 /CFT 2 as a test case to explore their properties. For both the AdS vacuum and multiboundary wormhole states, we find that the multiparty entanglement signals are generically non-zero and of order one in units of the central charge of the dual CFT, revealing substantial multiparty entanglement. In the large-horizon limit of multiboundary wormhole states, however, the signals for three or more parties become small, indicating the short-range nature of entanglement.
The AdS/CFT correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti-de Sitter space-time. We consider entangled states in the n-fold tensor product of a 1+1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is a black hole with n asymptotically AdS_3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. However, even when the bulk is completely connected, in some regions of the moduli space the entanglement is almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically n-partite entanglement, and use these to show that for some wormholes with n similar sized horizons there is intrinsic entanglement between at least n-1 parties, and that the distillable entanglement between the asymptotic regions is at least (n+1)/2 partite.
The AdS/CFT correspondence relates quantum entanglement between boundary conformal field theories and geometric connections in the dual asymptotically anti-de Sitter spacetime. We consider entangled states in the n-fold tensor product of a 1 + 1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is a black hole with n asymptotically AdS3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. However, even when the bulk is completely connected, there are regions of the moduli space in which the entanglement is instead almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically n-partite entanglement, and use these to show that for some wormholes with n similar sized horizons there is intrinsic entanglement between all n parties, and that the distillable entanglement between the asymptotic regions is at least (n + 1)/2 partite.
We show that in pure time-reflection-symmetric holographic states several known four-party entanglement signals vanish unless the triple information I3I_3I3​ is non-zero. In this sense, our results show that I3I_3I3​ is the strongest known signal of the presence of quadripartite entanglement. Additionally, I3I_3I3​ quantitatively bounds all four-party entanglement signals built from the multi-entropy. However, the residual entropy Q4Q_4Q4​, also a measure of four-party entanglement, is not bounded by I3I_3I3​, although I3=0I_3=0I3​=0 does imply Q4=0Q_4=0Q4​=0 for holographic states (except on a set of measure zero for which Q4Q_4Q4​ is ill-defined).
We study the real-time dynamics of multi-party entanglement signals in chaotic quantum many-body systems including but not necessarily restricted to holographic conformal field theories. We find that scrambling dynamics generates multiparty entanglement with rich structure including: (a) qualitatively different dynamical behaviours for different signals, likely reflecting different dynamics for different kinds of entanglement patterns, (b) discontinuities indicating dynamical phase transitions in the entanglement structure, (c) transient and non-monotonic multiparty entanglement, and (d) periods during which the extensive entanglement of some regions is entirely multipartite. Our main technical tool is the membrane theory of entanglement dynamics.
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.