Why measurements are made of effects
Tobias Fritz
TL;DR
This work addresses why measurements in physical theories are modeled as tuples of effects summing to a unit by introducing generalized measurement theories (GMTs) where measurements are primitive and states are derived. It demonstrates that in GMTs which are probabilistically separated, each measurement is representable as a normalized tuple of effects within an associated GPT of probabilistic states, providing a principled explanation for the prevalent effect-valued description in quantum theory. It further characterizes classical GMTs as precisely those arising from Boolean algebras, showing that strongly classical and projective GMTs correspond to $\mathcal{M}_{\mathcal{B}}$ and that projectivity discriminates between Boolean-algebra based classical measurements and quantum POVMs. Overall, the framework unifies classical, quantum, and more general measurement formalisms and clarifies when the effect-tuple description must hold, with implications for foundational questions in quantum theory and GPT reconstructions.
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.