The holographic dual of 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.

Islands and Uhlmann phase: explicit recovery of classical information from evaporating black holes

“…The entropy formula (1.1) is inspired by similar formulae in AdS/CFT and holography [16][17][18][19][20], but it is supposed to hold in a general theory of quantum gravity. Similarly, the protocol presented here takes inspiration from the recent identification of the holographic dual of the bulk symplectic form [21][22][23]. Suppose we are given a closed curve of holographic states |λ in AdS/CFT, and a boundary subregion of interest A.…”

“…This number depends only on the curve of reduced states ρ A -we will explain later the precise way in which it is defined. The key result of [23] was the formula…”

“…This is not a widely known object, so let us briefly review how it is defined. For a more extensive description see [24][25][26][27][28][29][30], or for the holographic context see [23].…”

Purification complexity without purifications

“…With the knowledge of the QFIM for any Gaussian states, it is interesting to consider the purification complexity for thermal states and mixed Gaussian states in a free QFT as what has been discussed in [37]. 12 Rather than assuming a free theory, the author in [62] calculated the Uhlmann's fidelity for two holographic states based on a replica trick. It would be intriguing to further investigate the QFIM and purification complexity based on that result.…”

“…Our proposal for the purification complexity C IM defined in (2.20) is based on the special properties of quantum Fisher information metric that is identified as Uhlmann's fidelity susceptibility. Recetly, different concepts associated with the geometry of quantum states also attract more attention in the field of AdS/CFT, see e.g., [62][63][64][65][66][67][68][69][70][71][72][73][74]. In [64] the authors consider the fidelity susceptibility (i.e., the Fubini-Study metric) of ground states in CFT from a small perturbation by a primary operator and argue its gravity dual is the volume of maximal time slice in an AdS spacetime.…”