On holographic time-like entanglement entropy

TL;DR

This paper extends holographic entanglement entropy to time-like regions by proposing a unique area prescription for mixed space-like and time-like extremal surfaces, addressing non-uniqueness issues.

Contribution

It introduces a new method to select a unique extremal surface area for time-like entanglement entropy holographically, improving previous approaches.

Findings

The complex area is generally non-unique for time-like extremal surfaces.

A new prescription selects a unique area from mixed extremal surfaces.

Examples confirm the validity of the proposed method.

Abstract

In order to study the pseudo entropy of time-like subregions holographically, the previous smooth space-like extremal surface was recently generalized to mix space-like and time-like segments and the area becomes complex value. This paper finds that, if one tries to use such kind of piecewise smooth extremal surfaces to compute time-like entanglement entropy holographically, the complex area is not unique in general. We then generalize the original holographic proposal of space-like entanglement entropy to pick up a unique area from all allowed ``space-like+time-like'' piecewise smooth extremal surfaces for a time-like subregion. We will give some concrete examples to show the correctness of our proposal.