Exact eigenvalue order statistics for the reduced density matrix of a bipartite system

Abstract

We consider the reduced density matrix $\rho_{A}^{(m)}$ of a bipartite system $AB$ of dimensionality $mn$ in a Gaussian ensemble of random, complex pure states of the composite system. For a given dimensionality $m$ of the subsystem $A$, the eigenvalues $\lambda_{1}^{(m)},\ldots, \lambda_{m}^{(m)}$ of $\rho_{A}^{(m)}$ are correlated random variables because their sum equals unity. The following quantities are known, among others: The joint probability density function (PDF) of the eigenvalues $\lambda_{1}^{(m)},\ldots, \lambda_{m}^{(m)}$ of $\rho_{A}^{(m)}$, the PDFs of the smallest eigenvalue $\lambda_{\rm min}^{(m)}$ and the largest eigenvalue $\lambda_{\rm max}^{(m)}$, and the family of average values $\langle \mathrm{Tr}\big(\rho_{A}^{(m)}\big)^{q}\rangle$ parametrised by $q$. Using values of $m$ running from $2$ to $6$ for definiteness, we show that these inputs suffice to identify and characterise the eigenvalue order statistics, i.e., to obtain explicit analytic expressions for the PDFs of each of the $m$ eigenvalues arranged in ascending order from the smallest to the largest one. When $m = n$ (respectively, $m < n$) these PDFs are polynomials of order $m^{2}-2$ (respectively, $mn - 2$) with support in specific sub-intervals of the unit interval, demarcated by appropriate unit step functions. Our exact results are fully corroborated by numerically generated histograms of the ordered set of eigenvalues corresponding to ensembles of over $10^{5}$ random complex pure states of the bipartite system. Finally, we present the general solution for arbitrary values of the subsystem dimensions $m$ and $n$, namely, formal exact expressions for the PDFs of every ordered eigenvalue.