Counterfactual distribution of cat states

In the counterfactual cryptography scheme proposed by Noh (2009), the sender Alice probabilistically transmits classical information to the receiver Bob without the physical travel of a particle. Here we generalize this idea to the distribution of quantum entanglement. The key insight is to replace their classical input choices with quantum superpositions. We further show that the scheme can be generalized to counterfactually distribute multi-partite cat states.