Explicit formulae for the Bures metric

The aim of this paper is to derive explicit formulae for the computation of the Riemannian Bures metric g on the manifold D of (finite-dimensional) no nsingular density matrices rho. This Riemannian metric introduced by Uhlman n generalizes the Fubini-Study metric to mixed states and is the infinitesi mal version of the Bures distance. Several formulae are known for computing the Bures metric in low dimensions. The formulae presented in this paper a llow for computing in finite dimensions without any diagonalization procedu res. The first equations we give are, essentially, of the form g(rho) = Sig ma a(ij) Tr d rho rho(j-1) d rho rho(j-1), where a(ij) is a matrix of invar iants of rho. A further formula, g(rho) = Sigma c(ij) dp(i) X dp(j) + Sigma b(ij) Tr d rho rho(i-1) d rho rho(j-1), is adapted to the local orthogonal decomposition D approximate to R-n X U(n)/T-n at generic points.