Holographic Pseudo Entropy

TL;DR

This paper introduces pseudo entropy, a generalization of entanglement entropy via post-selection, explores its properties in quantum systems and holography, and demonstrates its applications in various CFT contexts and gravity duals.

Contribution

It defines pseudo entropy, studies its properties in quantum and holographic systems, and proposes a gravity dual and a mixed state generalization.

Findings

Pseudo entropy reduces when local operators approach the subsystem boundary.

Holographic pseudo entropy matches perturbative calculations for Janus solutions.

Linearity property holds for holographic states, linking pseudo entropy to area operator weak values.

Abstract

We introduce a quantity, called pseudo entropy, as a generalization of entanglement entropy via post-selection. In the AdS/CFT correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo entropy for random states. We then calculate the pseudo entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo entropy and give its gravity dual.