The impossibility of simultaneous measurements implies irreversible disturbance when acting on physical systems

In 1927, Heisenberg put forward a heuristic argument to justify the existence of physical quantities that cannot be measured simultaneously with arbitrary precision. The argument, involving the famous gamma-ray microscope Gedankenexperiment, is based on the fact that acting on a quantum system irreversibly alters it. However, today, thanks to our modern understanding of quantum theory, we know that Heisenberg's argument is not precise. The impossibility of simultaneously measuring some physical quantities, termed measurement incompatibility, implies irreversible disturbance, namely the existence of operations that irreversibly alter the system on which they act. Not vice-versa. In our works, following an operational approach exploiting the framework of Operational Probabilistic Theories, we formally prove that measurement incompatibility implies irreversible disturbance and exhibit two counterexamples for the converse implication. In particular, these counterexamples are two toy theories: Minimal Classical Theory and Minimal Strongly Causal Bilocal Classical Theory. These two are distinct as counterexamples because the latter allows for classical conditioning. Furthermore, our toy theories also show that it is possible to devise classical theories that satisfy the properties of no-information without disturbance and no-broadcasting. The notion of classicality we adopt in our works is that the state spaces are simplexes whose pure states can be jointly perfectly discriminated. However, they are also Kochen-Specker and generalised noncontextual.