Geometry of quantum systems: density states and entanglement

Various problems concerning the geometry of the space u*(H) of Hermitian operators on a Hilbert space H are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kahler submanifolds. It is also shown that the space D(H) of density states on an n-dimensional Hilbert space H is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space D-k(H) of rank-k states, k = 1,..., n, is a smooth manifold of (real) dimension 2nk - k(2) - 1 and this stratification is maximal in the sense that every smooth curve in D(H), viewed as a subset of the dual u*(H) to the Lie algebra of the unitary group U (R), at every point must be tangent to the strata D-k(H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H-1 circle times H-2, an abstract criterion of entanglement is proved.