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arXiv:1807.04276

Entanglement Holonomies

Bartłomiej Czech, Lampros Lamprou, Leonard Susskind

Published Jul 11, 2018• 8 citations
arXivHTMLPDF

Abstract

We introduce a quantum notion of parallel transport between subsystems of a quantum state whose holonomies characterize the structure of entanglement. In AdS/CFT, entanglement holonomies are reflected in the bulk spacetime connection. When the subsystems are a pair of holographic CFTs in an entangled state, our quantum transport measures Wilson lines threading the dual wormhole. For subregions of a single CFT it is generated by the modular Berry connection and computes the effect of the AdS curvature on the transport of minimal surfaces. Our observation reveals a new aspect of the spacetime-entanglement duality and yet another concept shared between gravity and quantum mechanics.

DOI
10.48550/arxiv.1807.04276

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We introduce a quantum notion of parallel transport between subsystems of a quantum state whose holonomies characterize the structure of entanglement. In AdS/CFT, entanglement holonomies are reflected in the bulk spacetime connection. When the subsystems are a pair of holographic CFTs in an entangled state, our quantum transport measures Wilson lines threading the dual wormhole. For subregions of a single CFT it is generated by the modular Berry connection and computes the effect of the AdS curvature on the transport of minimal surfaces. Our observation reveals a new aspect of the spacetime-entanglement duality and yet another concept shared between gravity and quantum mechanics.
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.
A bstract We relate the Riemann curvature of a holographic spacetime to an entangle- ment property of the dual CFT state: the Berry curvature of its modular Hamiltonians. The modular Berry connection encodes the relative bases of nearby CFT subregions while its bulk dual, restricted to the code subspace, relates the edge-mode frames of the cor- responding entanglement wedges. At leading order in 1 /N and for sufficiently smooth HRRT surfaces, the modular Berry connection simply sews together the orthonormal co- ordinate systems covering neighborhoods of HRRT surfaces. This geometric perspective on entanglement is a promising new tool for connecting the dynamics of entanglement and gravitation.
Abstract: In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N, subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curvature is the symplectic form of the entanglement wedge. This generalises previous work on holographic Berry curvature.
A bstract In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N , subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curvature is the symplectic form of the entanglement wedge. This generalises previous work on holographic Berry curvature.
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 Berry connection describes transformations induced by adiabatically varying Hamiltonians. We study how zero modes of the modular Hamiltonian are affected by varying the region that supplies the modular Hamiltonian. In the vacuum of a 2D conformal field theory, global conformal symmetry singles out a unique modular Berry connection, which we compute directly and in the dual three-dimensional anti-de Sitter (AdS_{3}) picture. In certain cases, Wilson loops of the modular Berry connection compute lengths of curves in AdS_{3}, reproducing the differential entropy formula. Modular Berry transformations can be measured by bulk observers moving with varying accelerations.