Geometry of quantum systems: density states and entanglement

Abstract

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 Kähler 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 (H), 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 ⊗ H 2 , an abstract criterion of entanglement is proved.

On the Relation between States and Maps in Infinite Dimensions

“…Every quantum state admits such a realization and a reasoning analogous to the one in [6] shows that µ S is infinite-convex, non-negative and vanishes exactly on separable states.…”

Basics of Quantum Mechanics, Geometrization and Some Applications to Quantum Information

“…We shall call density states all convex combinations of pure states, and we denote them by D(H) [20,11].…”

Subnormalized states and trace-nonincreasing maps

“…Even after splitting off the N-phases of [U (1)] N , a residual arbitrariness still remains in the diagonalization of ρ, related to the fact that different permutations of N generically different eigenvalues Λ i belong to the same unitary orbit. This explains the factor N !…”

“…In the simplest case of one qubit the set M 2 is equivalent, with respect to the Hilbert-Schmidt (Euclidean) geometry, to a three dimensional ball, B 3 . For higher dimensions the geometry of M N gets more complicated and differs from the ball B N 2 −1 [1,2].…”