Classical theories, incompatibility and measurement disturbance

One of the peculiar aspects of quantum theory is the existence of pairs of observables that cannot be measured simultaneously. Extracting information about one of the elements of the pair causes an irreversible disturbance on the other, making it impossible to recover information about the value of the latter before the action on the system.

The notions of incompatibility and measurement disturbance have been recently formalised within the framework of operational probabilistic theories, generalising the quantum definition to generic information processing theories. This allowed us to prove in a theory-independent manner that these two properties are strictly related. Every operational theory that admits incompatible operations has some information that cannot be extracted without disturbance and vice versa. Furthermore, this has also allowed us to prove that contrarily to what one might expect, neither of them can be considered a signature of non-classicality. In particular, focusing on the study of operations whose input and output systems are non-trivial (i.e. not focusing on state preparations and measurements after which the physical system is discarded) two toy theories were built that are classical, posses incompatible operations, and satisfy no-information without disturbance. The first one is called Minimal Classical Theory (MCT). This is a subtheory of classical theory where the set of operations an experimenter is allowed to perform is restricted to the bones. Even the possibility of conditioning which operation to perform based on a previous one is removed, otherwise the full classical theory would be recovered. The second one is termed Minimal Strongly causal Bilocal Classical Theory (MSBCT) and was born to overcome the limitation of MCT where conditioning is forbidden. This theory represents a minimal version of a bilocal version of classical theory where all conditioned operations are allowed. The possible generalisations of the results obtained with these two toy theories to more general classes of operational theories and their consequences will also be discussed.