Simulability of observables in general probabilistic theories

TL;DR

This paper explores the simulation of observables in general probabilistic theories, introducing the concept of simulation irreducibility and analyzing how observables can be reconstructed from fundamental components, with implications for understanding measurement compatibility.

Contribution

It introduces the notion of simulation irreducibility in general probabilistic theories and characterizes how all observables can be simulated from these fundamental elements, extending compatibility concepts.

Findings

Simulation irreducible observables are fundamental for simulating any observable.

Quantum extreme rank-1 POVMs are the simulation irreducible observables.

Restrictions on simulators relate to compatibility and k-compatibility.

Abstract

The existence of incompatibility is one of the most fundamental features of quantum theory, and can be found at the core of many of the theory's distinguishing features, such as Bell inequality violations and the no-broadcasting theorem. A scheme for obtaining new observables from existing ones via classical operations, the so-called simulation of observables, has led to an extension of the notion of compatibility for measurements. We consider the simulation of observables within the operational framework of general probabilistic theories and introduce the concept of simulation irreducibility. While a simulation irreducible observable can only be simulated by itself, we show that any observable can be simulated by simulation irreducible observables, which in the quantum case correspond to extreme rank-1 positive-operator-valued measures. We also consider cases where the set of simulators is restricted in one of two ways: in terms of either the number of simulating observables or their number of outcomes. The former is seen to be closely connected to compatibility and $k$-compatibility, whereas the latter leads to a partial characterization for dichotomic observables. In addition to the quantum case, we further demonstrate these concepts in state spaces described by regular polygons.