Can imaginarity be broadcast via real operations?

Imaginarity has proven to be a valuable resource in various quantum information processing tasks.A natural question arises:can the imaginarity of quantum states be broadcast via real operations?In this work,we present explicit structures for nonreal states whose imaginarity can be broadcast and cloned.That is,for a nonreal state,its imaginarity can be cloned if and only if it is a direct sum of several maximally imaginary states under orthogonal transformation,and its imaginarity can be broadcast if and only if it is a direct sum of a real state and some nonreal qubit states which are mixtures of two orthogonal maximally imaginary states under orthogonal transformation.In particular,we show that for a nonreal pure state,its imaginarity cannot be broadcast unless it is a maximally imaginary state.Furthermore,we derive a trade-off relation on the imaginarity broadcasting of pure states in terms of the measure of irreversibility of quantum states concerning real operations and the geometric measure of imaginarity.In addition,we demonstrate that any faithful measure of imaginarity is not superadditive.