论文标题
Scott在广义概率理论中的连续性
Scott Continuity in Generalized Probabilistic Theories
论文作者
论文摘要
斯科特连续性是域理论的一个概念，在冯·诺伊曼代数理论中具有出乎意料的前世。斯科特连续状态称为正常状态，正常状态正是来自密度矩阵的状态。鉴于此，以及斯科特连续性在域理论中的有用性，自然要问这是否载入了广义概率理论。我们表明答案是否的 - 有无限维凸集集合，在相应的2值POVMS集合中，Scott连续状态的集合不会恢复原始凸组，但严格较大。这表明了在一般情况下使用拓扑来实现状态效应二元性的必要性，而不是纯粹是理论观念。
Scott continuity is a concept from domain theory that had an unexpected previous life in the theory of von Neumann algebras. Scott-continuous states are known as normal states, and normal states are exactly the states coming from density matrices. Given this, and the usefulness of Scott continuity in domain theory, it is natural to ask whether this carries over to generalized probabilistic theories. We show that the answer is no - there are infinite-dimensional convex sets for which the set of Scott-continuous states on the corresponding set of 2-valued POVMs does not recover the original convex set, but is strictly larger. This shows the necessity of the use of topologies for state-effect duality in the general case, rather than purely order theoretic notions.
