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Bures Metrics for Certain High-Dimensional Quantum Systems

Slater, P B

Published Jan 1, 1998• 2 citations
OpenAlex

Abstract

Hubner's formula for the Bures (statistical distance) metric is applied to both a one-parameter and a two-parameter series (n=2,...,7) of sets of 2^n x 2^n density matrices. In the doubly-parameterized series, the sets are comprised of the n-fold tensor products --- corresponding to n independent, identical quantum systems --- of the 2 x 2 density matrices with real entries. The Gaussian curvatures of the corresponding Bures metrics are found to be constants (4/n). In the second series of 2^n x 2^n density matrices studied, the singly-parameterized sets are formed --- following a study of Krattenthaler and Slater --- by averaging with respect to a certain Gibbs distribution, the n-fold tensor products of the 2 x 2 density matrices with complex entries. For n = 100, we are also able to compute the Bures distance between two arbitrary (not necessarily neighboring) density matrices in this particular series, making use of the eigenvalue formulas of Krattenthaler and Slater, together with the knowledge that the 2^n x 2^n density matrices in this series commute.

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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.

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Optics and Spectroscopy•Sep 1, 2007•6 citations
Hubner's formula for the Bures (statistical distance) metric is applied to both a one-parameter and a two-parameter series (n=2,...,7) of sets of 2^n x 2^n density matrices. In the doubly-parameterized series, the sets are comprised of the n-fold tensor products --- corresponding to n independent, identical quantum systems --- of the 2 x 2 density matrices with real entries. The Gaussian curvatures of the corresponding Bures metrics are found to be constants (4/n). In the second series of 2^n x 2^n density matrices studied, the singly-parameterized sets are formed --- following a study of Krattenthaler and Slater --- by averaging with respect to a certain Gibbs distribution, the n-fold tensor products of the 2 x 2 density matrices with complex entries. For n = 100, we are also able to compute the Bures distance between two arbitrary (not necessarily neighboring) density matrices in this particular series, making use of the eigenvalue formulas of Krattenthaler and Slater, together with the knowledge that the 2^n x 2^n density matrices in this series commute.
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.
An explicit parameterization for the state space of an nnn-level density matrix is given. The parameterization is based on the canonical coset decomposition of unitary matrices. We also compute, explicitly, the Bures metric tensor over the state space of two- and three-level quantum systems.
The aim of this paper is to derive explicit formulae for the computation of the Riemannian Bures metric g on the manifold of (finite-dimensional) nonsingular density matrices . This Riemannian metric introduced by Uhlmann generalizes the Fubini-Study metric to mixed states and is the infinitesimal version of the Bures distance. Several formulae are known for computing the Bures metric in low dimensions. The formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures. The first equations we give are, essentially, of the form , where is a matrix of invariants of . A further formula, , is adapted to the local orthogonal decomposition at generic points.
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.
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.