Extreme point of completely convex state structure

Abstract

It is well--known that the set of states of a given quantum mechanical system is to be closed from the point of view of the operational approach if we want to make mixed states or convex combinations. That is, $s_1$ and $s_2$ are states, then the same is to be true for $\lambda s_1 +(1-\lambda) s_2,$ where $0 < \lambda < 1.$ We can define a convex combination of elements in a linear space, but unfortunately, in the general case the linear space is artificial for the set of states and has no physical meaning, but the procedure of forming the mixtures of states has a natural meaning. This is why we provide an abstract definition of the mixtures, which is independent of the linearity notion. We call this space a convex structure.

In the work we consider state spaces, generalized state spaces, in which we select pure states, define operations and effects associated with the operations.

We also consider ultraproducts of the sequences of these structures, operations and
effects.