  We show that when non-commutative quantum mechanics is formulated on the
Hilbert space of Hilbert-Schmidt operators (referred to as quantum Hilbert
space) acting on a classical configuration space, spectral triplets as
introduced by Connes in the context of non-commutative geometry arise
naturally. A distance function as defined by Connes can therefore also be
introduced. We proceed to give a simple and general algorithm to compute this
function. Using this we compute the distance between pure and mixed states on
quantum Hilbert space and demonstrate a tantalizing link between statistics and
geometry.
