Covariant KSGNS construction and quantum instruments
Covariant KSGNS construction and quantum instruments
复制标题
协变 KSGNS 构造和量子仪器
DOI：
10.1142/s0129055x17500209
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J.-P. Pellonpaa
中科院分区：
文献类型：
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作者：
E.Haapasalo;J.-P. Pellonpaa
我们研究单位代数上的完全正 (CP) 倍半线性形式值映射，其中倍半线性形式在 a 代数上的模上运行。我们还研究了其中一个或两个代数都是冯诺依曼代数的情况。此外，我们假设 CP 映射对于对称群的作用是协变的。这使我们能够将这些图视为协变量子仪器的概括。我们确定协变 CP 图的最小协变膨胀（KSGNS 构造），以找到 CP 图在归一化协变 CP 图的凸子集中达到极端的必要和充分条件。作为一个特例，我们研究协变量子可观测量和仪器，其值空间是单模 I 型群的传递空间。最后，我们讨论关于平方可积表示协变的工具的情况。
We study completely positive (CP)-sesquilinear-form-valued maps on a unital-algebra, where the sesquilinear forms operate on a module over a-algebra. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.