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 invariants 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.

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Standard purification interlaces Hermitian and Riemannian metrics on the space of density operators with metrics and connections on the purifying Hilbert-Schmidt space. We discuss connections and metrics which are well adopted to purification, and present a selected set of relations between them. A connection, as well as a metric on state space, can be obtained from a metric on the purification space. We include a condition, with which this correspondence becomes one-to-one. Our methods are borrowed from elementary * -representation and fibre space theory. We lift, as an example, solutions of a von Neumann equation, write down holonomy invariants for cyclic ones, and "add noise" to a curve of pure states.

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The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Therefore, and also for mathematical reasons, it is natural to ask for curvature properties of this Riemannian metric. Here we determine its scalar curvature and Ricci tensor and prove a lower bound for the curvature on the submanifold of trace one matrices. This bound is achieved for the maximally mixed state, a further hint for the statistical meaning of the scalar curvature.