Necessity of macroscopic operation for the creation of superpositions of macroscopically distinct states

We consider the creation of superpositions of macroscopically distinct states by a completely-positive (CP) operation on a subsystem. We conclude that the subsystem on which the CP operation acts must be macroscopically large if the success probability of the CP operation does not vanish in the thermodynamic limit. In order to obtain this conclusion, we show two inequalities each of which represents a trade-off relation among the magnitude of an indicator for superpositions of macroscopically distinct states, the success probability of a CP operation, and the volume of the subsystem on which the CP operation acts.