双随机算子与量子随机变量
Bistochastic operators and quantum random variables
摘要：
给定一个作用在局部紧T2空间$X$的Borel集上的正算子值测度$ν$，其结果在（可能无限维）希尔伯特空间$\mathcal H$上的所有有界算子的代数$\mathcal B(\mathcal H)$中，人们可以考虑$ν$ -可积函数$X\rightarrow \mathcal B(\mathcal H)$，它们是正量子随机变量。我们定义了一个在商导致一个Banach空间的这些功能的跨度上的一个矩阵。我们考虑双随机算子作用在这个空间上，量子随机变量的优控制被定义为相对于这些算子。在经典的优控制理论，我们在这种情况下，涉及到一个不等式，涉及所有可能的凸函数的某种类型的优控制。与经典的设置不同，连续性和收敛性问题在整个工作中出现。
英文摘要：
Given a positive operator-valued measure $ν$ acting on the Borel sets of a locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional) Hilbert space $\mathcal H$, one can consider $ν$-integrable functions $X\rightarrow \mathcal B(\mathcal H)$ that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.
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