Abstract:
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.
Keywords:Hilbert space, linear operator, $\mathrm C^*$-algebra, von Neumann algebra, normed ideal operator space, quantum state, finitely additive measure, barycenter, extreme point, quantum correlations generated by a state.
The work of the second author was carried out within the framework of the Program for the Development of the Scientific and Educational Mathematical Center of the Volga Federal District (agreement no. 075-02-2023-944).
Citation:
G. G. Amosov, A. M. Bikchentaev, V. Zh. Sakbaev, “On Extreme Points of Sets in Operator Spaces and State Spaces”, Noncommutative Analysis and Quantum Information Theory, Collected papers. Dedicated to Academician Alexander Semenovich Holevo on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 324, Steklov Math. Inst., Moscow, 2024, 10–23; Proc. Steklov Inst. Math., 324 (2024), 4–17
\Bibitem{AmoBikSak24}
\by G.~G.~Amosov, A.~M.~Bikchentaev, V.~Zh.~Sakbaev
\paper On Extreme Points of Sets in Operator Spaces and State Spaces
\inbook Noncommutative Analysis and Quantum Information Theory
\bookinfo Collected papers. Dedicated to Academician Alexander Semenovich Holevo on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2024
\vol 324
\pages 10--23
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4376}
\crossref{https://doi.org/10.4213/tm4376}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4767943}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2024
\vol 324
\pages 4--17
\crossref{https://doi.org/10.1134/S0081543824010024}
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