Coarse–graining states on ultraproducts of probability algebras

Abstract

A probability algebra, which is a kind of analogue of a probability space, is designed to describe a physical system. If we have a prior distribution defined by a faithful state on this algebra and we can define the behavior of the system in terms of this state, then we can also make some prediction about its behavior. In practice, however, it is impossible to measure all observable quantities on the original algebra, so we only perform a partial measurement, that is, we measure the observable quantities on a certain subalgebra. This measurement then determines another state on this subalgebra. If the resulting state extends to the original algebra without introducing additional information about the behavior of system, then we call the resulting state on the original algebra coarse–graining.

In the paper we discuss the existence of coarse–graining states, referred to as $(\mathcal {B}, \omega)$–coarse–graining of a state, and we also consider ultraproducts of sequences of probability algebras and states. We show that under the introduced definition, ultraproducts of sequences of probability algebras are probability algebras, and we discuss the existence of conditional mathematical expectations, coarse–grained states, and information.