Incompatibility of observables, channels and instruments in information theories

Every theory of information, including classical and quantum, can be studied in the framework operational probabilistic theories--where notion test generalizes that quantum instrument, namely a collection operations summing to channel, simple rules are given for composition tests parallel sequence. Here we study compatibility an theory. Following literature, first introduce strong compatibility, then illustrate its ultimate relaxation, deem weak compatibility. It is shown two notions coincide case observation tests--which counterpart POVMs--while there exist weakly compatible channels not strongly compatible. We prove necessary sufficient conditions exhibit incompatible tests. show admits if only some information cannot extracted without disturbance.