Minimal Areas from Entangled Matrices

TL;DR

This paper introduces a new way to define and compute entanglement entropy in matrix quantum mechanics, linking it to minimal boundary areas similar to holographic principles, and emphasizing the role of coarse-graining.

Contribution

It presents a relational subsystem framework in matrix quantum mechanics that relates entanglement entropy to minimal boundary areas, inspired by the Ryu-Takayanagi formula.

Findings

Entanglement entropy can be expressed as a minimization problem over boundary areas.

The construction connects entanglement, noncommutative geometry, and quantum reference frames.

Coarse-graining is crucial to manage non-geometric subregions in the sum.

Abstract

We define a relational notion of a subsystem in theories of matrix quantum mechanics and show how the corresponding entanglement entropy can be given as a minimisation, exhibiting many similarities to the Ryu-Takayanagi formula. Our construction brings together the physics of entanglement edge modes, noncommutative geometry and quantum internal reference frames, to define a subsystem whose reduced state is (approximately) an incoherent sum of density matrices, corresponding to distinct spatial subregions. We show that in states where geometry emerges from semiclassical matrices, this sum is dominated by the subregion with minimal boundary area. As in the Ryu-Takayanagi formula, it is the computation of the entanglement that determines the subregion. We find that coarse-graining is essential in our microscopic derivation, in order to control the proliferation of highly curved and disconnected non-geometric subregions in the sum.