A necessary condition for incompatibility of observables in general probabilistic theories

We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply 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 illustrate our findings with a number of examples including the introduced notion of reverse observables.