Matrix Quantum Mechanics and Entanglement Entropy: A Review

TL;DR

This review explores entanglement entropy in matrix quantum mechanics, its relation to string theory and non-commutative geometry, and how large N limits connect to holographic principles like the Ryu-Takayanagi formula.

Contribution

It synthesizes recent developments on entanglement in MQM, especially the connection to area laws and holography, highlighting conditions for minimal area entanglement in large N gauge theories.

Findings

Target space entanglement can obey an area law at large N.

U(N) invariance relates entanglement entropy to minimal surface areas.

Examples demonstrate entanglement entropy calculations in non-commutative geometries.

Abstract

We review aspects of entanglement entropy in the quantum mechanics of $N\times N$ 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.