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arXiv:quant-ph/9810012 · Journal of Geometry and Physics

The scalar curvature of the Bures metric on the space of density matrices

Jochen Dittmann

Published Aug 1, 1999• 21 citations
arXivHTMLPDF
DOI
10.1016/s0393-0440(98)00068-0

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Bures Metrics for Certain High-Dimensional Quantum Systems

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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.

•Jan 1, 1998•2 citations
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.