Categorical characterizations of operator-valued measures

TL;DR

This paper explores the mathematical foundations of quantum measurements modeled by POVMs, establishing dualities and equivalences between different categorical and algebraic representations, including von Neumann algebras.

Contribution

It introduces a categorical duality framework for POVMs and shows their equivalence to morphisms between von Neumann algebras in the continuous case.

Findings

Establishes a duality between categories of POVMs and effect algebras

Shows equivalence of POVMs with morphisms between von Neumann algebras for continuous cases

Provides a unified categorical perspective on quantum measurement models

Abstract

The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras.