Dilation Theory

Abstract

Naimark’s work in the 1940s on the representation of positive operator measures in terms of projection valued measures acting in a larger Hilbert space may be seen as the starting point for the discovery of a vast variety of dilation theorems, an early highlight being Stinespring’s dilation theorem from the mid 1950s. This chapter contains a unified approach to the sort of dilation theorems that are particulary useful for quantum mechanics. Besides the Naimark and Stinespring results, these include an analogous two-variable theory and the so-called Kraus representation in a general functional analytic setting. One also encounters the first inklings of these results in action: the sections on operations and instruments and measurement dilations point to the central quantum mechanical applications which are the main theme of this book.