Typical Entanglement

TL;DR

This paper derives the typical eigenvalue distribution of reduced density matrices for random pure states in high-dimensional Hilbert spaces using a statistical mechanics approach, revealing insights into bipartite entanglement.

Contribution

It introduces a novel saddle point and Coulomb gas method to analytically determine the typical spectrum of reduced density matrices for random states.

Findings

Derived the eigenvalue distribution for typical reduced density matrices.

Analyzed cases of unbiased ensembles and fixed purity.

Applied a statistical mechanics framework to quantum entanglement analysis.

Abstract

Let a pure state \psi be chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix \rho of an N-dimensional subsystem. The bipartite entanglement properties of \psi are encoded in the spectrum of \rho. By means of a saddle point method and using a "Coulomb gas" model for the eigenvalues, we obtain the typical spectrum of reduced density matrices. We consider the cases of an unbiased ensemble of pure states and of a fixed value of the purity. We finally obtain the eigenvalue distribution by using a statistical mechanics approach based on the introduction of a partition function.