Information and disturbance in a physical theory

Alessandro Tosini (Universita di Pavia), Paolo Perinotti (Universita di Pavia), Giacomo Mauro D’Ariano (Universita di Pavia)

Any experiment is intended to provide information on a system via a measurement. However, as we learn from quantum theory, it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. The interplay between information and disturbance has been largely investigated as a discriminating feature between classical and quantum physics and the no-information without disturbance theorem has been proved to be a quantum feature.
The quantum scenario leads to an operational interpretation of the information-disturbance re- lation that is deeply anchored to other features of quantum theory, as the tomography of states via local measurements, the purificability of mixed state and the causal structure. In particular, the traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local tomography.

Here we introduce a notion of disturbance that holds for an arbitrary theory. Then we prove that a system satisfies no-information without disturbance if and only if the identical evolution cannot result form the coarse-graining of more elementary operations. We also provide other suf- ficient conditions for no-information without disturbance. Finally, we prove a structure theorem probabilistic theories showing that any system decomposes in a classical system and a system that satisfies no-information without disturbance. Via concrete examples we exhibit the independence of no-information without disturbance from other typically quantum features as local tomography and states purification.