Relation between time- and spacelike entanglement entropy

TL;DR

This paper uncovers a fundamental connection between timelike and spacelike entanglement entropy, showing that the former can be derived from the latter and its temporal derivative, with implications for causality and quantum field theory.

Contribution

It introduces a novel relation linking timelike and spacelike entanglement entropy, applicable to a broad class of states and constructed perturbatively for more general cases.

Findings

Timelike entanglement entropy is related to spacelike entanglement entropy and its derivative.

The relation holds exactly for states conformally equivalent to the vacuum in 2D CFTs.

For general states, the relation can be constructed perturbatively.

Abstract

In this study, we establish a connection between timelike and spacelike entanglement entropy. We show that timelike entanglement entropy is closely related to spacelike entanglement entropy and its temporal derivative. For a broad class of states, it can be uniquely determined by a linear combination of spacelike entanglement entropy and its first-order temporal derivative. This relation holds, for instance, in states conformally equivalent to the vacuum in two-dimensional conformal field theories. For more general states, we demonstrate that the relation can be constructed perturbatively. Our results suggest that timelike entanglement entropy is constrained by causality. Moreover, this relation provides a unified framework for timelike and spacelike entanglement entropy, within which the imaginary component of timelike entanglement entropy can be understood as arising from the non-commutativity between the twist operator and its first-order temporal derivative.