Why measurements are made of effects

TL;DR

This paper introduces generalized measurement theories to explore why quantum and probabilistic measurements are modeled as effects summing to the unit, providing a framework to question and relax this assumption.

Contribution

It develops a new mathematical framework called generalized measurement theories (GMTs) and characterizes when measurements are effects, offering insights into the classicality of theories.

Findings

Measurements are made of effects in GMTs where probabilistic states separate measurements.

The separation condition for states and measurements is physically well-motivated.

GMTs corresponding to Boolean algebras are strongly classical and projective.

Abstract

Both in quantum theory and in general probabilistic theories, measurements with $n$ outcomes are modelled as $n$-tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.