A ug 1 99 8 Note on Explicit Formulae for the Bures Metric

The aim of this paper is to derive explicit formulae for the Riemannian Bures metric g on the manifold D of (finite dimensional) nondegenerate density matrices ̺. The computation of the Bures metric using these equations does not require any diagonalization procedure. The first equations we give are, essentially , of the form g = a ij Tr d̺ ̺ i−1 d̺ ̺ j−1 , where a i,j is a matrix of invari-ants of ̺. A further formula, g = c ij dp i ⊗ dp j + b ij Tr d̺ ̺ i−1 d̺ ̺ j−1 , is adapted to the local orthogonal decomposition D ≈ R n × U(n) / T n at generic points.