量子論における同時操作不可能性とその順序構造による特徴づけ

我们研究了X×X上的正核，其中X是一个具有群的作用的集合，并在C*-代数a上的一个（不一定是Hilbert）模块上的a -半线性形式的集合中取值。这些映射被假设是关于X上的群作用的协变的，并且是群在可逆（a -线性）模块映射集合中的表示。在正协变核的凸子集上，我们得到了这类核的极值性的充分必要条件。我们的重点主要放在这些核的一个特殊例子上：一个完全正（CP）协变映射，我们获得了协变最小膨胀（或KSGNS构造）。我们确定了归一化协变CP映射集合的极值点，并作为一个特例，研究了值空间为非模i型群的传递空间的协变量子观测和仪器。作为一个例子，我们讨论了相对于平方可积表示的协变仪器的情况。

We study positive kernels on X×X, where X is a set equipped with an action of a group, and taking values in the set of A-sesquilinear forms on a (not necessarily Hilbert) module over a C*-algebra A. These maps are assumed to be covariant with respect to the group action on X and a representation of the group in the set of invertible (A-linear) module maps. We find necessary and sufficient conditions for extremality of such kernels in certain convex subsets of positive covariant kernels. Our focus is mainly on a particular example of these kernels: a completely positive (CP) covariant map for which we obtain a covariant minimal dilation (or KSGNS construction). We determine the extreme points of the set of normalized covariant CP maps and, as a special case, study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. As an example, we discuss the case of instruments that are covariant with respect to a square-integrable representation.