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arXiv:2205.01723

The 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 May 3, 2022• 0 citations
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Abstract

We present marginal cumulative distribution functions (CDF) for density matrices $ρ$ of fixed purity $\tfrac{1}{N}\leμ_N(ρ)=\textrm{Tr}[ρ^2]\le 1$ 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 $μ_N(ρ)\in[\tfrac{1}{N}, 1]$. 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 $μ_4(ρ)\in[\tfrac{1}{4}, 1]$ 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 $ρ$ for fixed purity at very high dimension are briefly discussed.

DOI
10.48550/arxiv.2205.01723

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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.
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 study 12 parameter families of two qubit density matrices, arising from a special class of two-fermion systems with four single particle states or alternatively from a four-qubit state with amplitudes arranged in an antisymmetric matrix. We calculate the Wooters concurrences and the negativities in a closed form and study their behavior. We use these results to show that the relevant entanglement measures satisfy the generalized Coffman-Kundu-Wootters formula of distributed entanglement. An explicit formula for the residual tangle is also given. The geometry of such density matrices is elaborated in some detail. In particular an explicit form for the Bures metric is given.
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. The required observables only exist if it is possible to identify d mutually unbiased (MU) complex (d×d) Hadamard matrices, defining a complete set of d+1 MU bases. This thesis is an exploration of sets of r≤d+1 MU bases in low dimensions. We derive all inequivalent sets of MU bases in dimensions two to five confirming that in these dimensions, the complete sets of (d+1) MU bases are unique. In dimension six, we prescribe a first Hadamard matrix and construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and never find more than two that are mutually unbiased. We also study subsets of a complete set of MU bases by considering sets of pure states which satisfy the desired properties. We use this concept to provide the strongest numerical evidence so far that no seven MU bases exist in dimension six. In the final part of the thesis, we introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack is higher than the BB84 or six-state protocols when the legitimate parties use a complete set of MU bases.
The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.