Target space entanglement in quantum mechanics of fermions and matrices

TL;DR

This paper studies entanglement properties of fermions in matrix models, deriving bounds and scaling laws for entanglement entropy and mutual information, confirming an area law in the large N limit.

Contribution

It introduces an algebraic approach to target space entanglement in fermionic matrix models and provides analytical results for entropy bounds and scaling behaviors.

Findings

Upper bounds of Renyi entropies are N log 2 for N particles.

Single-interval entropy scales as (1/3) log N in the large N limit.

Analytical expression for mutual information between two intervals at large N.

Abstract

We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general R\'enyi entropies are $N \log 2$ for $N$ particles or an $N\times N$ matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as $\frac{1}{3}\log N$ in the large $N$ model. We obtain an analytical $\mathcal{O}(N^0)$ expression of the mutual information for two intervals in the large $N$ expansion.