The Monge metric on the sphere and geometry of quantum states

Topological and geometrical properties of the set of mixed quantum states i n the N-dimensional Hilbert space are analysed. Assuming that the correspon ding classical dynamics takes place on the sphere we use the vector SU(2) c oherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a si mple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state p(*), proporti onal to the identity matrix, admits the largest value for the coherent stat es, while the delocalized 'chaotic' states are close to p(*). This contrast s the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) me trics, for which the distance from p(*) is the same for all pure states. We discuss possible physical consequences including unitary time evolution an d the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.