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The existence of incompatibility is one of the most fundamental features ofquantum theory, and can be found at the core of many of the theory'sdistinguishing features, such as Bell inequality violations and theno-broadcasting theorem. A scheme for obtaining new observables from existingones via classical operations, the so-called simulation of observables, has ledto an extension of the notion of compatibility for measurements. We considerthe simulation of observables within the operational framework of generalprobabilistic theories and introduce the concept of simulation irreducibility.While a simulation irreducible observable can only be simulated by itself, weshow that any observable can be simulated by simulation irreducibleobservables, which in the quantum case correspond to extreme rank-1positive-operator-valued measures. We also consider cases where the set ofsimulators is restricted in one of two ways: in terms of either the number ofsimulating observables or their number of outcomes. The former is seen to beclosely connected to compatibility and $k$-compatibility, whereas the latterleads to a partial characterization for dichotomic observables. In addition tothe quantum case, we further demonstrate these concepts in state spacesdescribed by regular polygons.

Simulability of observables in general probabilistic theories