On the extreme points of sets in spaces of operators and spaces of states

Abstract

A representation of the set of quantum states by barycenters of non-negative normalized finite additive measures on a unit sphere of Hilbert space H is obtained. In terms of the properties of a measure on a unit sphere of space H, the conditions for its barycenter to belong to the set of extreme points of the set of quantum states and to the set of normal states are derived. A characterization of the unitary elements of the unital C∗-algebra in terms of extreme points is obtained. The extreme points of the unit ball E1 of the normalized ideal space of operators ⟨E,∥⋅∥E⟩ on H are investigated. If U∈extr(E1) for some unitary operator U∈B(H), then V∈extr(E1) for all unitary operators V∈B(H). Quantum correlations corresponding to the singular state on the algebra of all bounded operators in a Hilbert space are constructed.