The Holographic Dual of the Entanglement Wedge Symplectic Form

TL;DR

This paper establishes a boundary dual for the bulk symplectic form in holography using Uhlmann holonomy, connecting quantum information concepts with geometric structures in the entanglement wedge.

Contribution

It introduces a novel boundary dual of the entanglement wedge symplectic form via Uhlmann holonomy, extending holographic Berry curvature studies.

Findings

Uhlmann holonomy computed for boundary subregion states.

Berry phase expressed as integral of an abelian connection.

Curvature of this connection equals the entanglement wedge symplectic form.

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.