Simulability of observables in general probabilistic theories

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.

Figure 1

Simulation of a new observable by using observables $\mathsf{A}, \mathsf{B}$, and $\mathsf{C}$. In every individual measurement, the random number generator (RNG) chooses one of observables $\mathsf{A}, \mathsf{B}$, and $\mathsf{C}$. Then, the classical outcome is affected by a classical channel $\nu$.

Figure 2

Simulation can be equivalently seen as a mixture of postprocessed observables.

Figure 3

Multiple uses of a same observables are allowed but do not alter the generality of simulability. The scheme on the left-hand side can be reduced to the scheme on the right-hand side.

Figure 4

The set of simulable observables is convex.

Figure 5

The set of simulable observables is closed under postprocessings.

Figure 6

The unbiased effects in $\mathfrak{sim}(\mathsf{X},\mathsf{Y},\mathsf{Z})$ form an octahedron.

Figure 7

Square state space (above) and the space of effects (below).

Figure 8

Odd and even polygon state spaces $\mathcal{S}$ and corresponding sets of effects $\mathcal{E}\left(\mathcal{S}\right)$.