Distribution of density matrices at fixed purity for arbitrary dimensions

Abstract

Uniformly sampling quantum density matrices has been demonstrated for a long time [Mezzadri, Notices AMS 54, 592 (2007), \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski, J. Math. Phys. 52, 062201 (2011)] and is a valuable tool in the analysis and testing of quantum relations (e.g., the tightness of an uncertainty relation); in ranking the performance of various tools to witness entanglement, nonlocality, discord, etc.; and in probing the geometry of fundamental quantum properties (e.g, the boundary between separable and entangled states). While uniformly sampling quantum states has been shown to be quite valuable, there is to our knowledge no such method to sample these states uniformly at a constant amount of mixedness. This challenge becomes increasingly acute as post-selecting a subensemble of states within a narrow range of, say, purities from this fully uniform sampling becomes overwhelmingly inefficient at high dimension. In this work, we demonstrate how to analytically construct the distribution of density matrices at constant purity (the trace of the square of a density matrix) as our mixedness measure. With this technique at our disposal: we provide analytic formulas for the cumulative radial distribution functions for the uniform distribution of density matrices at constant purity for arbitrary total dimension $N$, with closed form expressions up to $N=4$, and perform numerical integration to analyze the statistics of these uniform-purity distributions at higher dimensions (up to $N=100$). Moreover, we use our new capabilities to: compare different measures of entanglement and nonclassicality, including Wootter's concurrence, the logarithmic negativity and quantum discord, as a function of purity at dimension $N=4$ (two qubits); to find the probability distribution of purities of marginal density matrices from a uniform distribution of joint pure states (and thus the entanglement statistics of this uniform ensemble); and to numerically test a conjectured uncertainty relation between quantum mutual information and the sum of classical mutual informations in complementary bases.