Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

Understand

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues.

• For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint.
• We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N.
• Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman.