# Matrix Quantum Mechanics and Entanglement Entropy: A Review

## TL;DR

This paper reviews how entanglement entropy behaves in matrix quantum mechanics, especially at large N, and connects it to string theory and non-commutative geometries.

## Contribution

The paper provides a comprehensive review of entanglement entropy in matrix quantum mechanics and its connections to string theory and non-commutative geometries.

## Key findings

- Entanglement entropy in matrix quantum mechanics displays an area law at large N under certain conditions.
- U(N) invariance in matrix quantum mechanics implies a minimal area formula for entanglement entropy.
- Target space entanglement is reviewed in the context of non-commutative field theories.

## Abstract

We review aspects of entanglement entropy in the quantum mechanics of N×N matrices, i.e., matrix quantum mechanics (MQM), at large N. In doing so, we review standard models of MQM and their relation to string theory, D-brane physics, and emergent non-commutative geometries. We overview, in generality, definitions of subsystems and entanglement entropies in theories with gauge redundancy and discuss the additional structure required for definining subsystems in MQMs possessing a U(N) gauge redundancy. In connecting these subsystems to non-commutative geometry, we review several works on ‘target space entanglement,’ and entanglement in non-commutative field theories, highlighting the conditions in which target space entanglement entropy displays an ‘area law’ at large N. We summarize several example calculations of entanglement entropy in non-commutative geometries and MQMs. We review recent work in connecting the area law entanglement of MQM to the Ryu–Takayanagi formula, highlighting the conditions in which U(N) invariance implies a minimal area formula for the entanglement entropy at large N. Finally, we make comments on open questions and research directions.