**Information measure of counterfactuality in communication protocols**

The classical theories of communication rely on the assumption that there has to be a flow of particles from Bob to Alice in order for him to send a message to her. We have recently developed a quantum protocol [1] that allows Alice to perceive Bob's message “counterfactually” - that is, without Alice receiving any particles that have interacted with Bob. By utilising a scheme built on results from interaction-free measurements and the quantum Zeno effect, we have proposed a protocol in which information travels in the opposite direction to the emitted particles. In comparison to previous attempts on such protocols, this one is such that a weak interaction at Bob's laboratory would not leave a weak trace [2] that could be detected by Alice.

We proceed the study of our counterfactual communication scheme by adapting methods of phase estimations in quantum optics [3] to a framework of parameter estimation in Mach-Zehnder interferometer structures. We use two information measures: the Shannon mutual information and the classical Fisher information. We show how a weak interaction with the particle in Bob's laboratory gives rise to Fisher information. The Fisher information is shown to be proportional to the integrated wavefunction squared at the point of interaction. Whence we argue that the Fisher information is a reasonable measure of the extent to which Alice's particle was present in Bob's laboratory.

Some schemes for counterfactual communication and counterfactual transmission of quantum states have been based on nested Mach-Zehnder interferometers (NMZIs).[4] These schemes do not allow for particles to propagate from Alice to Bob or vice versa. We can, thus, use the Fisher information to provide a methodology for evaluating the counterfactuality of them. Our results show that the widely discussed NMZI protocol of Salih et al. [4], only exhibits counterfactuality with perfect quantum channels. For realistic devices this protocol is no more counterfactual than a direct transmission between source and receiver. Our protocol, however, allows particles to be transmitted in the opposite direction of the message, which avoids the counterfactual violation, even in real non-ideal devices.

[1] D. R. M. Arvidsson-Shukur and C. H. W. Barnes, Phys. Rev. A 94, 062303 (2016).

[2] L. Vaidman, Phys. Rev. A 87, 052104 (2013).

[3] T. B. Bahder, Phys. Rev. A 83, 053601 (2011).

[4] H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, Phys. Rev. Lett. 110, 170502 (2013).