  A protocol for transmission of information between two parties introduced by
Salih et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves
sending quantum amplitude back and forth through a quantum channel in a series
of steps, rather than simply sending a signal in one direction. The authors
claimed that their protocol was ``counterfactual'' in the sense that while a
quantum channel is needed to connect the parties, its actual usage becomes
vanishingly small in the asymptotic limit as the number of steps tends to
infinity. Here we show that this claim is incorrect because it uses
probabilistic reasoning that is not valid at intermediate times in the presence
of quantum interference. When ill-defined probabilities are replaced with a
well-defined measure of channel usage here called ``Cost'', equal to the
absolute square of the amplitude sent through the channel, the total Cost does
not go to zero in the asymptotic limit of a large number of steps, but is
bounded below by a rigorous inequality. A detailed analysis shows that this
bound is satisfied in the SLAZ protocol. The analysis leading to the bound uses
the fact that the Gram matrix formed by inner products of a collection of pure
quantum states is additive over Hilbert subspaces and invariant under unitary
time transformations. Its off-diagonal elements, which in general are not
positive, play a significant role in the formal argument as well as providing a
somewhat strange way of visualizing the transfer of information.
