We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.
We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.
Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024), but expressed surprise when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) uses the minimum number of parameters possible, N^2 - 1, as opposed to N^2 + N - 1 (although this over-parameterization, as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities, and the need for additional (possibly supercomputer) analyses is indicated (particularly so, for higher-dimensional quantum systems, such as the 2 x 3 we also study here).
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices $\ensuremath{\varrho}$ describing N-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for $N=2\ifmmode\times\else\texttimes\fi{}2$ and $N=2\ifmmode\times\else\texttimes\fi{}3$ cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.
We pursue a number of analytical directions, motivated to some extent initially by the possibility of developing a methodology for formally proving or disproving a certain conjecture of quantum-theoretical relevance (quant-ph/0308037). It asserts that the 15-dimensional volume occupied by the separable two-qubit density matrices is (\sqrt{2}-1)/3, as measured in terms of the statistical distinguishability metric (four times the Bures or minimal monotone metric). Somewhat disappointingly, however, the several various analyses that we report, though we hope of independent/autonomous interest, appear to provide small indication of how to definitively resolve the conjecture. Among our studies here are ones of: (1) the Bures volumes of the two-dimensional sections of Bloch vectors for a number of the Jakobczyk-Siennicki two-qubit scenarios; (2) the structure of certain convex polytopes of separable density matrices; and (3) the diagonalization of 15 x 15 Bures metric tensors.
Probability theory is a measure theory – with a soul. Mark Kac Section 2.6 was devoted to classical ensembles , that is to say ensembles defined by probability measures on the set of classical probability distributions over N events. In this chapter quantum ensembles are defined by choosing probability measures on the set of density matrices of size N . A warning should be issued first: there is no single, naturally distinguished measure in M ( N ) , so we have to analyse several measures, each of them with different physical motivations, advantages and drawbacks. This is in contrast to the set of pure quantum states, where the Fubini–Study measure is the only natural choice for a measure that defines ‘random states’. A simple way to define a probability measure goes through a metric. Hence we will start this chapter with a review of the metrics defined on the set of mixed quantum states. Monotone metrics In Section 2.5 we explained how the Fisher metric holds its distinguished position due to the theorem of Čencov, which states that the Fisher metric is the unique monotone metric on the probability simplex Δ N –1 . Now that the latter has been replaced by the space of quantum states M ( N ) we must look again at the question of metrics. Since the uniqueness in the classical case came from the behaviour under stochastic maps, we turn our attention to stochastic quantum maps – the completely positive, trace preserving maps discussed in Chapter 10. A distance D in the space of quantum states M ( N ) is called monotone if it does not grow under the action of a stochastic map Φ, If a monotone distance is geodesic the corresponding metric on M ( N ) is called monotone.
We derive explicit expressions for the volume elements of both the minimal and maximal monotone metrics over the (n^{2} - 1)-dimensional convex set of n x n density matrices for the cases n = 3 and 4. We make further progress for the specific n = 3 maximal-monotone case, by taking the limit of a certain ratio of integration results, obtained using an orthogonal set of eight coordinates. By doing so, we find remarkably simple marginal probability distributions based on the corresponding volume element, which we then use for thermodynamic purposes. We, thus, find a spin-1 analogue of the Langevin function. In the fully general n = 4 situation, however, we are impeded in making similar progress by the inability to diagonalize a 3 x 3 Hermitian matrix and thereby obtain an orthogonal set of coordinates to use in the requisite integrations.