Multitime quantum communication: Interesting but not counterfactual

A protocol for transmission of information between two parties introduced by Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)] 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 protocol of Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)]. 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.