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arXiv:quant-ph/9808044 · Journal of Physics A Mathematical and General

Explicit formulae for the Bures metric

Jochen Dittmann

Published Jan 1, 1999• 95 citations
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Abstract

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.

DOI
10.1088/0305-4470/32/14/007

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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 consider the Riemannian geometry of the space of nonsingular density matrices D 1 equipped with the Bures metric g B .This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states.We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of (D 1 , g B ), which allow us to calculate other curvature quantities .It turns out that D 1 is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest.Moreover, we give a local description of D 1 and explicit formulae for the Bures metric in terms of natural matrix operations containing and d only.
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 compute the volume of the N^2-1 dimensional set M_N of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N^2-1 dimensional hyper-halfsphere of radius 1/2. For N=2 we obtain the volume of the Uhlmann 3-D hemisphere, embedded in R^4. We find also the area of the boundary of the set M_N and obtain analogous results for the smaller set of all real, symmetric density matrices. An explicit formula for the Bures-Hall normalization constants is derived for an arbitrary N.
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.
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 .