Information and disturbance in operational probabilistic theories. (arXiv:1907.07043v1 [quant-ph])

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.

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