Causality & holographic entanglement entropy

TL;DR

This paper establishes causality conditions for entanglement entropy in quantum field theories and proves that the holographic prescription for entanglement entropy respects these conditions when the bulk satisfies the null energy condition, introducing the entanglement wedge as a key concept.

Contribution

It demonstrates that the covariant holographic entanglement entropy prescription is consistent with causality constraints, introducing the entanglement wedge as the natural bulk region for a boundary state.

Findings

Holographic entanglement entropy obeys causality conditions under null energy condition

Introduction of the entanglement wedge as the bulk region associated with boundary states

Validation of the holographic prescription's consistency with relativistic causality

Abstract

We identify conditions for the entanglement entropy as a function of spatial region to be compatible with causality in an arbitrary relativistic quantum field theory. We then prove that the covariant holographic entanglement entropy prescription (which relates entanglement entropy of a given spatial region on the boundary to the area of a certain extremal surface in the bulk) obeys these conditions, as long as the bulk obeys the null energy condition. While necessary for the validity of the prescription, this consistency requirement is quite nontrivial from the bulk standpoint, and therefore provides important additional evidence for the prescription. In the process, we introduce a codimension-zero bulk region, named the entanglement wedge, naturally associated with the given boundary spatial region. We propose that the entanglement wedge is the most natural bulk region corresponding to the boundary reduced density matrix.