Modular Flow as a Disentangler

TL;DR

This paper introduces a boundary-based measure called modular minimal entropy to analyze when holographic entanglement surfaces are in the same Cauchy slice, revealing insights into bulk causality and tensor network structures.

Contribution

It proposes a new boundary quantity, the modular minimal entropy, to determine the relative positioning of HRT surfaces and explore bulk causality in holography.

Findings

Modular minimal entropy bounds the constrained surface area.

It indicates whether HRT surfaces are in the same Cauchy slice.

The formula is proven in 2D or with local modular Hamiltonians.

Abstract

In holographic duality, the entanglement entropy of a boundary region is proposed to be dual to the area of an extremal codimension-2 surface that is homologous to the boundary region, known as the Hubeny-Rangamani-Takayanagi (HRT) surface. In this paper, we study when the HRT surfaces of two boundary subregions R, A are in the same Cauchy slice. This condition is necessary for the subregion-subregion mapping to be local for both subregions and for states to have a tensor network description. To quantify this, we study the area of a surface that is homologous to A and is extremal except at possible intersections with the HRT surface of R (minimizing over all such possible surfaces), which we call the constrained area. We give a boundary proposal for an upper bound of this quantity, a bound which is saturated when the constrained surface intersects the HRT surface of R at a constant angle. This boundary quantity is the minimum entropy of region A in a modular evolved state -- a state that has been evolved unitarily with the modular Hamiltonian of R. We can prove this formula in two boundary dimensions or when the modular Hamiltonian is local. This modular minimal entropy is a boundary quantity that probes bulk causality and, from this quantity, we can extract whether two HRT surfaces are in the future or past of each other. These entropies satisfy some inequalities reminiscent of strong subadditivity and can be used to remove certain corner divergences.