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arXiv:math-ph/0102032

Bures geometry of the three-level quantum systems. II

Paul B. Slater

Published Feb 26, 2001• 0 citations
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Abstract

For the eight-dimensional Riemannian manifold comprised by the three-level quantum systems endowed with the Bures metric, we numerically approximate the integrals over the manifold of several functions of the curvature and of its (anti-)self-dual parts. The motivation for pursuing this research is to elaborate upon the findings of Dittmann in his paper, "Yang-Mills equation and Bures metric" (quant-ph/9806018).

DOI
10.48550/arxiv.math-ph/0102032

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For the eight-dimensional Riemannian manifold comprised by the three-level quantum systems endowed with the Bures metric, we numerically approximate the integrals over the manifold of several functions of the curvature and of its (anti-)self-dual parts. The motivation for pursuing this research is to elaborate upon the findings of Dittmann in his paper, "Yang-Mills equation and Bures metric" (quant-ph/9806018).
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Utilizing a number of results of Dittmann, we investigate the nature of the Yang-Mills field over the eight-dimensional convex set, endowed with the Bures metric, of three-level quantum systems. Adopting a numerical strategy, we first decompose the field into self-dual and anti-self-dual components, by implementing the octonionic equations of Corrigan, Devchand, Fairlie and Nuyts. For each of these three fields, we obtain approximations to: (1) the Yang-Mills functional; (2) certain quantities studied by Bilge, Dereli, and Kocak in their analysis of self-dual Yang-Mills fields in eight dimensions; and (3) other measures of interest.
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.
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.
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.