Connecting Fisher information to bulk entanglement in holography

Abstract

In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.

v3: 28 pages, 1 figure. Accepted for publication in JHEP. Major update: added section 2.2 which discusses our proposal for states under scalar perturbations as well. Conclusions unchanged. Typos corrected, references added