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Physical Review Research

Distribution of density matrices at fixed purity for arbitrary dimensions

Paul M. Alsing, Christopher C. Tison, James Schneeloch, Richard J. Birrittella, Michael L. Fanto

Published Nov 17, 2022• 4 citations
Publisher

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.

DOI
10.1103/physrevresearch.4.043114

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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 NNN, with closed form expressions up to N=4N=4N=4, and perform numerical integration to analyze the statistics of these uniform-purity distributions at higher dimensions (up to N=100N=100N=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=4N=4N=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.
We present marginal cumulative distribution functions (CDF) for density matrices ρρρ of fixed purity 1N≤μN(ρ)=Tr[ρ2]≤1\tfrac{1}{N}\leμ_N(ρ)=\textrm{Tr}[ρ^2]\le 1N1​≤μN​(ρ)=Tr[ρ2]≤1 for arbitrary dimension NNN. We give closed form analytic formulas for the cases N=2N=2N=2 (trivial), N=3N=3N=3 and N=4N=4N=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 μN(ρ)∈[1N,1]μ_N(ρ)\in[\tfrac{1}{N}, 1]μN​(ρ)∈[N1​,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 μ4(ρ)∈[14,1]μ_4(ρ)\in[\tfrac{1}{4}, 1]μ4​(ρ)∈[41​,1] for the case of N=4N=4N=4 (two qubits). We also investigate the distribution of eigenvalues of a reduced NNN-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 ρρρ for fixed purity at very high dimension are briefly discussed.
One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density matrices are mixtures of dyads, where a dyad has the form uu' for any any unit column vector u. These unit vectors are the elementary events of the generalized probability space. Perhaps the simplest case to see that something unusual is going on is the case of uniform density matrix, i.e. 1/n times identity. This matrix assigns probability 1/n to every unit vector, but of course there are infinitely many of them. The new normalization rule thus says that sum of probabilities over any orthonormal basis of directions is one. We develop a probability calculus based on these more general distributions that includes definitions of joints, conditionals and formulas that relate these, i.e. analogs of the theorem of total probability, various Bayes rules for the calculation of posterior density matrices, etc. The resulting calculus parallels the familiar 'classical' probability calculus and always retains the latter as a special case when all matrices are diagonal. Whereas the classical Bayesian methods maintain uncertainty about which model is 'best', the generalization maintains uncertainty about which unit direction has the largest variance. Surprisingly the bounds also generalize: as in the classical setting we bound the negative log likelihood of the data by the negative log likelihood of the MAP estimator.
The local purity of large many-body quantum systems can be studied by following a statistical mechanical approach based on a random matrix model. Restricting the analysis to the case of global pure states, this method proved to be successful, and a full characterization of the statistical properties of the local purity was obtained by computing the partition function of the problem. Here we generalize these techniques to the case of global mixed states. In this context, by uniformly sampling the phase space of states with assigned global mixedness, we determine the exact expression of the first two moments of the local purity and a general expression for the moments of higher order. This generalizes previous results obtained for globally pure configurations. Furthermore, through the introduction of a partition function for a suitable canonical ensemble, we compute the approximate expression of the first moment of the marginal purity in the high-temperature regime. In the process, we establish a formal connection with the theory of quantum twirling maps that provides an alternative, possibly fruitful, way of performing the calculation.
To estimate the degree of quantum entanglement of random pure states, it is crucial to understand the statistical behavior of entanglement indicators such as the von Neumann entropy, quantum purity, and entanglement capacity. These entanglement metrics are functions of the spectrum of density matrices, and their statistical behavior over different generic state ensembles have been intensively studied in the literature. As an alternative metric, in this work, we study the sum of the square root spectrum of density matrices, which is relevant to negativity and fidelity in quantum information processing. In particular, we derive the finite-size mean and variance formulas of the sum of the square root spectrum over the Bures-Hall ensemble, extending known results obtained recently over the Hilbert-Schmidt ensemble.
We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
Investigation about various random quantum state ensembles is always focused on its eigenvalue statistics. Analytical derivation of joint probability distribution density functions of quantum state ensembles can be used to study asymptotical phenomenon and concentration of measure phenomenon in high dimensions. In this paper, however, we will derive the spectral density of equiprobable mixture of two random density matrices for qubits. Besides, we will also work out the spectral density of mixture under the so-called quantum addition rule (see details in the text), proposed in this http URL (J.Math.Phys. 57, 052202 (2016)] of two random density matrices for qubits. We also introduce a new divergence by mimicking the quantum Jensen-Shannon divergence. Unfortunately, our numerical experiments show that both the divergence and its square root are not valid metric.