Smallest eigenvalue distribution of the fixed trace Laguerre beta-ensemble

TL;DR

This paper derives the distribution of the smallest eigenvalue of the fixed trace Laguerre beta-ensemble, revealing that in the large N limit, it coincides with the classical Laguerre ensemble, thus solving an open problem in quantum entanglement.

Contribution

It provides an exact expression for the smallest eigenvalue distribution using Jack polynomials and shows the large N limit matches the classical Laguerre ensemble, addressing an open problem.

Findings

Exact distribution expressed via Jack polynomials for finite N.

Large N limit matches classical Laguerre ensemble distribution.

Global fixed trace constraint does not affect local eigenvalue correlations in large N limit.

Abstract

In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre ensemble of $N\times N$ random matrices. We showed that for finite $N$ the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found, a limiting distribution, when the smallest eigenvalue is suitably scaled with $N$ followed by a large $N$ limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large $N$ limit. Consequently, we have solved an open problem: The determination of the smallest eigenvalue distribution of the reduced density matrix---obtained by tracing out the environmental degrees of freedom---for a bipartite quantum system of unequal dimensions.