scholarly journals Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices

Bernoulli ◽  
2012 ◽  
Vol 18 (1) ◽  
pp. 322-359 ◽  
Author(s):  
Zongming Ma
Keyword(s):  
Wishart Matrices ◽  
Extreme Eigenvalues

scholarly journals Eigenvalue distributions of beta-Wishart matrices

2014 ◽  
Vol 03 (02) ◽  
pp. 1450009 ◽  
Author(s):  
Alan Edelman ◽  
Plamen Koev
Keyword(s):  
Hypergeometric Function ◽  
Random Matrices ◽  
The Real ◽  
Wishart Matrices ◽  
Extreme Eigenvalues ◽  
Matrix Argument ◽  
Explicit Expressions

We derive explicit expressions for the distributions of the extreme eigenvalues of the beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0.

scholarly journals Extreme eigenvalues of Wishart matrices: application to entangled bipartite system

2018 ◽  
Author(s):  
Carlo W. J. Beenakker
Keyword(s):  
Hilbert Spaces ◽  
Matrix Theory ◽  
Minimum Eigenvalue ◽  
Bipartite Entanglement ◽  
Wishart Matrices ◽  
Extreme Eigenvalues ◽  
Bipartite System ◽  
System P ◽  
Random Pure State ◽  
Reduced Density

This article describes the application of random matrix theory (RMT) to the estimation of the bipartite entanglement of a quantum system, with particular emphasis on the extreme eigenvalues of Wishart matrices. It first provides an overview of some spectral properties of unconstrained Wishart matrices before introducing the problem of the random pure state of an entangled quantum bipartite system consisting of two subsystems whose Hilbert spaces have dimensions M and N respectively with N ≤ M. The focus is on the smallest eigenvalue which serves as an important measure of entanglement between the two subsystems. The minimum eigenvalue distribution for quadratic matrices is also considered. The article shows that the N eigenvalues of the reduced density matrix of the smaller subsystem are distributed exactly as the eigenvalues of a Wishart matrix, except that the eigenvalues satisfy a global constraint: the trace is fixed to be unity.

Sign in / Sign up

Export Citation Format

Share Document