Entanglement renormalization and integral geometry

TL;DR

This paper explores the connection between integral geometry and entanglement in AdS$_3$, showing how entanglement contours relate to the metric of kinematic space and extending the framework to higher dimensions.

Contribution

It introduces a novel interpretation of the kinematic space metric as the entanglement contour and generalizes integral geometric methods to higher-dimensional bulk spaces.

Findings

Entanglement contour can be realized as the metric of kinematic space.

Holographic understanding of disentangler and isometry operations.

Derived a renormalization group equation for long-distance entanglement contour.

Abstract

We revisit the applications of integral geometry in AdS$_3$ and argue that the metric of the kinematic space can be realized as the entanglement contour, which is defined as the additive entanglement density. From the renormalization of the entanglement contour, we can holographically understand the operations of disentangler and isometry in multi-scale entanglement renormalization ansatz. Furthermore, a renormalization group equation of the long-distance entanglement contour is then derived. We then generalize this integral geometric construction to higher dimensions and in particular demonstrate how it works in bulk space of homogeneity and isotropy.