On the States Having Pure-State Restrictions for a Pair of Regions with a Non-Trivial Intersection

Abstract

For given two regions of the lattice, assume that our state has a pure-state restriction for each of them. We discuss whether its restrictions to the intersection and the union regions are pure states or not. It is immediate to see that the answer is affirmative for any tensor-product systems. (We, however, include its proof for the completeness sake.) In this note, we consider mostly CAR systems. We show that the assertion above holds for any such a state of finite-dimensional CAR systems. For infinite-dimensional CAR systems, assuming additionally the product properties between each of the regions and its complement region on the states satisfying the standing assumption, we can show the purity on the intersection region for them. However, the latter part of the assertion will not always hold for the infinite-dimensional CAR systems unlike for the finite-dimensional CAR or any dimensional tensor-product systems. We establish the criterions for the purity and the non-purity on the union region for those states of the infinite-dimensional CAR systems.