Developing Teiko Heinosaari's ideas, we have quantified the intrinsic noise content of an observable in a general probabilistic theory and derived a noise content inequality for incompatible observables. We have applied the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We have illustrated our findings with a number of examples including the introduced notion of reverse observables.
S. N. Filippov, T. Heinosaari, L. Leppajarvi. Necessary condition for incompatibility of observables in general probabilistic theories. Phys. Rev. A 95, 032127 (2017), E-print arXiv:1609.08416 [quant-ph]