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

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