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Optics and Spectroscopy

Coset parameterization of density matrices

Seyed Javad Akhtarshenas

Published Sep 1, 2007• 6 citations
Publisher

Abstract

We give an explicit parameterization for the state space of an n -level density matrix. The parameterization is based on the canonical coset decomposition of unitary matrices. As an application, we also calculate explicitly the Bures metric tensor over the state space of two-level quantum system.
DOI
10.1134/s0030400x0709010x

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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.