Operational restrictions in general probabilistic theories

TL;DR

This paper investigates how operational restrictions on measurements in general probabilistic theories affect the structure and capabilities of physical systems, including quantum mechanics, by classifying and characterizing different types of measurement limitations.

Contribution

It introduces a framework for understanding operational measurement restrictions, classifies them into three types, and fully characterizes restrictions based on effects within general probabilistic theories.

Findings

Restrictions from effects can be fully characterized

Severe limitations can occur even with all effects accessible

Physically meaningful restrictions exist beyond effect-based limitations

Abstract

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular.