Finitely Correlated Pure States

Abstract

This paper is a report on joint work with Mark Fannes and Bruno Nachtergaele on a class of states on one-dimensional quantum spin systems, called “finitely correlated”. These states can be constructed quite explicitly, which in itself is remarkable, since apart from quasi-free states or convex combinations of product states, there seem to be no states allowing a similarly complete control. One of their characteristic properties is the absence of a certain, typically quantum mechanical, “frustration” phenomenon. In this sense they do not exhibit the full complexity of quantum mechanical correla tions. On the other hand, the structure of this class is very rich, and it is large enough to approximate any translationally invariant state (in the weak*-topology). Moreover, finitely correlated states appear as the exact ground states in physical models, particu larly of antiferromagnetism. Therefore their study is an ideal vantage point from which to begin an exploration of the complexities of quantum mechanical correlations. The aim of the present article is to give an overview of some of the results obtained so far, emphasizing motivations and intuitive ideas, and referring to the journal articles for the technical aspects, and further references.