The distribution of density matrices at fixed purity for arbitrary dimensions

Abstract

We present marginal cumulative distribution functions (CDF) for density matrices $\rho$ of fixed purity $\tfrac{1}{N}\le\mu_N(\rho)=\textrm{Tr}[\rho^2]\le 1$ for arbitrary dimension $N$. We give closed form analytic formulas for the cases $N=2$ (trivial), $N=3$ and $N=4$, and present a prescription for CDFs of higher arbitrary dimensions. These formulas allows one to uniformly sample density matrices at a user selected, fixed constant purity, and also detail how these density matrices are distributed nonlinearly in the range $\mu_N(\rho)\in[\tfrac{1}{N}, 1]$. As an illustration of these formulas, we compare the logarithmic negativity and quantum discord to the (Wootter's) concurrence spanning a range of fixed purity values in $\mu_4(\rho)\in[\tfrac{1}{4}, 1]$ for the case of $N=4$ (two qubits). We also investigate the distribution of eigenvalues of a reduced $N$-dimensional obtained by tracing out the reservoir of its higher-dimensional purification. Lastly, we numerically investigate a recently proposed complementary-quantum correlation conjecture which lower bounds the quantum mutual information of a bipartite system by the sum of classical mutual informations obtained from two pairs of mutually unbiased measurements. Finally, numerical implementation issues for the computation of the CDFs and inverse CDFs necessary for uniform sampling $\rho$ for fixed purity at very high dimension are briefly discussed.