Information and disturbance in operational probabilistic theories

Giacomo Mauro D'Ariano, Paolo Perinotti, Alessandro Tosini

Published 2019-07-16Version 1

Any measurement is intended to provide information on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. 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 discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. We then prove an equivalent condition for no-information without disturbance-atomicity of the identity-namely the impossibility of achieving the trivial evolution-the identity-as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. As a consequence we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. Besides proving that no-information without disturbance is implied by the purification postulate, we show via concrete examples that the converse is not true. Finally we show that no-information without disturbance and local discriminability are independent.