Connections and metrics respecting purification of quantum states

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 fiber 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.