Smallest eigenvalue distribution of the fixed trace Laguerre beta-ensemble

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.