We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parametrizations of Boya et al (1998 Preprint quant-ph/9810084) of the n × n density matrices, in terms of squared components of the unit ( n - 1)-sphere and the n × n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, thereby obtaining `Bures prior probability distributions' over the two- and three-state systems. Then, as a first step in extending these results to n >3, we determine that the `Hall normalization constant' ( C n ) for the marginal Bures prior probablity distribution over the ( n - 1)-dimensional simplex of the n eigenvalues of the n × n density matrices is, for n = 4, equal to 71 680/ 2 . Since we also find that C 3 = 35/ , it follows that C 4 is simply equal to 2 11 C 3 / . ( C 2 itself is known to equal 2/ .) The constant C 5 is also found. It too is associated with a remarkably simple decomposition, involving the product of the eight consecutive prime numbers from 3 to 23. We also preliminarily investigate several cases n >5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n . This would have diverse applications, including quantum inference and universal quantum coding.
We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parametrizations of Boya et al (1998 Preprint quant-ph/9810084) of the n × n density matrices, in terms of squared components of the unit ( n - 1)-sphere and the n × n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, thereby obtaining Bures prior probability distributions' over the two- and three-state systems. Then, as a first step in extending these results to n >3, we determine that the Hall normalization constant' ( C n ) for the marginal Bures prior probablity distribution over the ( n - 1)-dimensional simplex of the n eigenvalues of the n × n density matrices is, for n = 4, equal to 71 680/ 2 . Since we also find that C 3 = 35/ , it follows that C 4 is simply equal to 2 11 C 3 / . ( C 2 itself is known to equal 2/ .) The constant C 5 is also found. It too is associated with a remarkably simple decomposition, involving the product of the eight consecutive prime numbers from 3 to 23. We also preliminarily investigate several cases n >5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n . This would have diverse applications, including quantum inference and universal quantum coding.
We apply and extend recent results of Krattenthaler and Slater (quant-ph/9612043), who sought quantum analogs of seminal work on universal data compression of Clarke and Barron. KS obtained explicit formulas for the eigenvalues and eigenvectors of the 2^m x 2^m density matrices gotten by averaging the m-fold tensor products with themselves of the 2 x 2 density matrices. The weighting was done with respect to a one-parameter family of probability distributions, all the members of which are spherically-symmetric over the "Bloch sphere" of two-level quantum systems. This family includes the normalized volume element of the minimal monotone (Bures) metric. In this letter, we conduct parallel analyses (for m =2,3,4), based on a natural measure on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera and Lewenstein (quant-ph/9804024) and find interesting similarities and differences with the findings of KS. In addition, we are able to obtain exact analogous results, based on the measure of ZHSL, for the twofold tensor products of the 3 x 3 density matrices.
We extend to additional probability measures and scenarios, certain of the recent results of Krattenthaler and Slater (quant-ph/9612043), whose original motivation was to obtain quantum analogs of seminal work on universal data compression of Clarke and Barron. KS obtained explicit formulas for the eigenvalues and eigenvectors of the 2^m x 2^m density matrices derived by averaging the m-fold tensor products with themselves of the 2 x 2 density matrices. The weighting was done with respect to a one-parameter (u) family of probability distributions, all the members of which are spherically-symmetric (SU(2)-invariant) over the ``Bloch sphere'' of two-level quantum systems. For u = 1/2, one obtains the normalized volume element of the minimal monotone (Bures) metric. In this paper, analyses parallel to those of KS are conducted, based on an alternative "natural" measure on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera, and Lewenstein (quant-ph/9804024). The approaches of KS and that based on ZHSL are found to yield [1 + m/2] identical SU(2) x S_{m}-invariant eigenspaces (but not coincident eigenvalues for m > 3). Companion results, based on the SU(3) form of the ZHSL measure, are obtained for the twofold and threefold tensor products of the 3 x 3 density matrices. We find a rather remarkable limiting procedure (selection rule) for recovering from these analyses, the (permutationally-symmetrized) multiplets of SU(3) constructed from two or three particles. We also analyze the scenarios (all for m = 2) N = 2 x 3, N= 2 x 3 x 2, N= 3 x 2 x 2 and N = 4 and, in addition, generalize the ZHSL measure, so that it incorporates a family of (symmetric) Dirichlet distributions (rather than simply the uniform distribution), defined on the (N-1)-dimensional simplex of eigenvalues.
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.
In a previous study (quant-ph/0207181), we formulated a conjecture that arbitrarily coupled qubits (describable by 4 x 4 density matrices) are separable with an a priori probability of 8/(11 π^2) = 0.0736881. For this purpose, we employed the normalized volume element of the Bures (minimal monotone) metric as a probability distribution over the fifteen-dimensional convex set of 4 x 4 density matrices. Here, we provide further/independent (quasi-Monte Carlo numerical integration) evidence of a stronger nature (giving an estimate of 0.0736858 vs. 0.0737012 previously) for this conjecture. Additionally, employing a certain ansatz, we estimate the probabilities of separability based on certain other monotone metrics of interest. However, we find ourselves, at this point, unable to convincingly conjecture exact simple formulas for these new (smaller) probabilities.
We compute the volume of the ( N 2 − 1)-dimensional set N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an ( N 2 − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½ S 3 ⊂ 4 . We find also the area of the boundary of the set N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N .
In a number of previous studies, we have investigated the use of the volume element of the Bures (minimal monotone) metric -- identically, one-fourth of the statistical distinguishability (SD) metric -- as a natural measure over the (n^2-1)-dimensional convex set of n x n density matrices. This has led us for the cases n = 4 and 6 to estimates of the prior (Bures/SD) probabilities that qubit-qubit and qubit-qutrit pairs are separable. Here, we extend this work from such bipartite systems to the tripartite "laboratory'' quantum systems possessing U x U x U symmetry recently constructed by Eggeling and Werner (Phys. Rev. A 63 [2001], 042324). We derive the associated SD metric tensors for the three-qubit and three-qutrit cases, and then obtain estimates of the various related Bures/SD probabilities using Monte Carlo methods.