Modification of "Counterfactual communication protocols" which eliminates weak particle traces

Possibility to communicate between spatially separated regions, without even a single photon passing between the two parties, is an amazing quantum phenomenon. The possibility of transmitting one value of a bit in such a way, the interaction-free measurement, was known for quarter of a century. The protocols of full communication, including transmitting unknown quantum states were proposed only few years ago, but it was shown that in all these protocols the particle was leaving a weak trace in the transmission channel, the trace larger than the trace left by a single particle passing through the channel. This made the claim of counterfactuality of these protocols at best controversial. However, a simple modification of these recent protocols eliminates the trace in the transmission channel making all these protocols counterfactual.