Entangled random pure states with orthogonal symmetry: exact results

TL;DR

This paper provides exact analytical formulas for the distribution of Schmidt eigenvalues and the average Rényi entropy of entangled random pure states with orthogonal symmetry, applicable to arbitrary Hilbert space dimensions.

Contribution

It introduces new exact results for the density of Schmidt eigenvalues and average Rényi entropy for orthogonal symmetry states, extending previous work to arbitrary dimensions.

Findings

Analytical expression for Schmidt eigenvalue density

Exact average Rényi entropy for orthogonal symmetry states

Results agree well with numerical simulations

Abstract

We compute analytically the density $\varrho_{N,M}(\lambda)$ of Schmidt eigenvalues, distributed according to a fixed-trace Wishart-Laguerre measure, and the average R\'enyi entropy $\langle\mathcal{S}_q\rangle$ for reduced density matrices of entangled random pure states with orthogonal symmetry $(\beta=1)$. The results are valid for arbitrary dimensions $N=2k,M$ of the corresponding Hilbert space partitions, and are in excellent agreement with numerical simulations.