On Extreme Points of Sets in Operator Spaces and State Spaces

We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere $S_1(\mathcal H)$ of a Hilbert space $\mathcal H$. For a measure on $S_1(\mathcal H)$, we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital $\mathrm C^*$-algebra are characterized in terms of extreme points. We also study extreme points $\mathrm{extr}(\mathcal E^1)$ of the unit ball $\mathcal E^1$ of a normed ideal operator space $\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle$ on $\mathcal H$. If $U\in\mathrm{extr}(\mathcal E^1)$ for some unitary operator $U\in\mathcal{B}(\mathcal H)$, then $V\in\mathrm{extr}(\mathcal E^1)$ for all unitary operators $V\in\mathcal{B}(\mathcal H)$. In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.