On Extreme Points of Sets in Operator Spaces and State Spaces

Pub Date : 2024-07-11 DOI:10.1134/s0081543824010024

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

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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 \(\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle\) 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.

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论算子空间和状态空间中集合的极值点

摘要我们用希尔伯特空间 \(\mathcal H\) 的单位球 \(S_1(\mathcal H)\)上的非负归一化有限相加度量的arycenters 来表示量子态集合。对于 \(S_1(\mathcal H)\)上的一个度量，我们从它的性质方面找到条件，在这些条件下，这个度量的原心属于量子态族的极值点集合和正常态集合。我们用极值点来表征一个单素（\mathrm C^*\）代数的单元元。我们还研究了规范理想算子空间 \(\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle\) 上单位球 \(\mathrm{extr}(\mathcal E^1)\)的极值点。如果 \(U\in\mathrm{extr}(\mathcal E^1)\)对于某个单元算子 \(U\in\mathcal{B}(\mathcal H)\)，那么 \(V\in\mathrm{extr}(\mathcal E^1)\)对于所有单元算子 \(V\in\mathcal{B}(\mathcal H)\)。此外，我们还构建了与希尔伯特空间中所有有界算子代数上的奇异状态相对应的量子相关性。

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