POVMs and the Two Theorems of Naimark and Sz.-Nagy

TL;DR

This paper revisits Naimark and Sz.-Nagy theorems on quantum observables, providing an elementary proof and exploring implications for quantum statistical inference and the classical-quantum distinction.

Contribution

It offers a new elementary proof of Sz.-Nagy's theorem and extends the resolution to respect all products of observables, with implications for quantum inference.

Findings

Elementary proof of Sz.-Nagy's theorem using inner product spaces

Resolution respecting all observable products is established

Implications for classical statistical methods in quantum inference

Abstract

In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.