完全正算子的积分理论：李雅普诺夫型定理及应用
Integration Theory for Completely positive Instruments: A Lyapunov-Type Theorem and Applications
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中文总结 AI 辅助
该研究发展了量子完全正算子的积分理论，建立了CP映射值的李雅普诺夫定理，扩展了相关工作并建立了算子间的对应关系，连接了CP算子与POVMs积分理论。
中文摘要 AI 辅助
我们通过两种互补方法发展了关于量子算子的积分理论：一种是基于Bartle积分理论的向量测度公式，另一种是张量积构造。作为主要应用，我们建立了完全正（CP）映射值的李雅普诺夫定理，通过相关积分映射刻画非原子完全正算子值域的凸性。这将Plosker和Ramsey的工作从POVMs扩展到完全正算子，并对他们研究场景中的现象给出了一般性刻画。我们还建立了完全正算子与完全正映射之间的对应关系，证明了单位完全正算子的C*-凸集的Krein-Milman型定理，并表明张量积构造将CP算子理论与Farenick等人提出的POVMs积分理论相联系。
英文摘要
We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).
