Strength of interaction for information distribution

Let us consider two quantum systems: system A and system B. Suppose that a classical information is encoded to quantum states of the system A and we distribute this information to both systems by making them interact with each other. We show that it is impossible to achieve this goal perfectly if the strength of interaction between the quantum systems is smaller than a quantity that is determined by noncommutativity between a Hamiltonian of the system A and the states (density operators) used for the information encoding. It is a consequence of a generalized Winger-Araki-Yanase theorem which enables us to treat conserved quantities other than additive ones.