Counterfactual exchange of unknown quantum states

We show that two unknown quantum states, either pure or mixed, can be bidirectionally ``teleported'' counterfactually. In the process of information exchange, there is no physical particle traveling between two communicators. Unlike conventional quantum teleportation, our protocol requires neither prearranged entangled pairs nor Bell measurements. We entangle and disentangle a photon and an atom through multiple measurements in a special sequence. These multiple measurements can smoothly but reversibly control system evolution. The total effect of multiple measurements can be considered as a unitary time evolution operator. Therefore, our protocol can be regarded as a counterfactual quantum swap gate. Based on our approach, all known conventional quantum gates can be counterfactualized.