Geometry of quantum systems: density states and entanglement

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