Geometry of quantum systems: density states and entanglement

Abstract

Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ 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 $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ 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 $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.