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 K¨ahler 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 Dk(H) of rank-k states, k = 1, . . . , n, is a smooth manifold of (real) dimension 2nk − k2 − 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(H), at every point must be tangent to the strata Dk(H) it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition H = H1 ⊗ H2, an abstract criterion of entanglement is proved.