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

Abstract

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. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.

25 pag., 4 fig - minor changes, typos fixed. To appear in JSTAT

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• Subsystem dynamics under random Hamiltonian evolution
• Bures-Hall Ensemble: Spectral Densities and Average Entropies
• Recursion scheme for the largest $\beta$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport
• The difference between two random mixed quantum states: exact and asymptotic spectral analysis
• Recursion for the smallest eigenvalue density of $\beta$-Wishart-Laguerre ensemble
• Smallest eigenvalue density for regular or fixed-trace complex Wishart-Laguerre ensemble and entanglement in coupled kicked tops
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles
• Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model
• Supersymmetry Method for Chiral Random Matrix Theory with Arbitrary Rotation Invariant Weights
• Violations of Bell inequalities from random pure states
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• Distribution of Schmidt-like eigenvalues for Gaussian Ensembles of the Random Matrix Theory