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) nonsingular density matrices rho. 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 g(rho) = Sigma a(ij) Tr d rho rho(j-1) d rho rho(j-1), where a(ij) is a matrix of invariants 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.