Timelike entanglement entropy

TL;DR

This paper introduces the timelike entanglement entropy, a complex-valued measure of quantum information that extends entanglement concepts into the time direction, with implications for holography and emergent spacetime.

Contribution

It defines and computes the timelike EE in 2d theories, relates it to pseudo entropy, and explores its holographic dual involving complex extremal surfaces, advancing understanding of time in quantum entanglement.

Findings

Timelike EE is a complex-valued measure related to pseudo entropy.

Numerical and analytic results agree for 2d CFTs.

Holographic dual involves complex extremal surfaces.

Abstract

We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.