On Extreme Points of Sets in Operator Spaces and State Spaces

G. G. Amosov ; A. M. Bikchentaev ; V. Zh. Sakbaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 10-23

Résumé

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,\| \cdot \|_{\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.

Détail

Keywords: Hilbert space, linear operator, $\mathrm C^*$-algebra, von Neumann algebra, normed ideal operator space, quantum state, finitely additive measure, extreme point, quantum correlations generated by a state.
Mots-clés : barycenter