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.
The hierarchy of equations for reduced density matrices that describes a thermodynamically equilibrium quantum system obtained earlier by the author is investigated in the momentum representation. In the paper it is shown that the use of the momentum representation opens up new opportunities in studies of macroscopic quantum systems both nonsuperfluid and superfluid. It is found that the distribution over momenta in a quantum fluid is not a Bose or Fermi distribution even in the limit of practically noninteracting particles. The distribution looks like a Maxwellian one although, strictly speaking, it is not Maxwellian. The momentum distribution in a quantum crystal depends upon the interaction potential and the crystalline structure. The momentum distribution in a superfluid contains a delta function. The momentum distribution for the condensate in a superfluid crystal consists of delta peaks that are arranged periodically in momentum space. The periodical structure remains if the condensate crystal is not superfluid.
Given a uniform ensemble of quantum density matrices
ρ, it is useful to calculate the mean value over this ensemble of a product of entries of
ρ. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions
P=(P1,P2,...,PN) where the
Pj are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of
P components. The answer to the classical problem follows from an integral formula due to Dirichlet.