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arXiv:2107.05055 · Physical Review A

Three approaches for analyzing the counterfactuality of counterfactual protocols

Alon Wander, Eliahu Cohen, Lev Vaidman A

Published Jul 21, 2021• 11 citations
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Abstract

Counterfactual communication protocols are analyzed using three approaches: a classical argument, the weak trace criterion, and the Fisher information criterion. It is argued that the classical analysis leads to contradiction and should therefore be abandoned. The weak trace and Fisher information criteria are shown to agree about the degree of counterfactuality of communication protocols involving postselection. It is argued that postselection is a necessary ingredient of counterfactual communication protocols. Coherent interaction experiments, as well as a recently introduced modification of counterfactual communication setups which eliminates the weak trace, are discussed.

DOI
10.1103/physreva.104.012610

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Counterfactual communication protocols are analyzed using three approaches: a classical argument, the weak trace criterion, and the Fisher information criterion. It is argued that the classical analysis leads to contradiction and should therefore be abandoned. The weak trace and Fisher information criteria are shown to agree about the degree of counterfactuality of communication protocols involving postselection. It is argued that postselection is a necessary ingredient of counterfactual communication protocols. Coherent interaction experiments, as well as a recently introduced modification of counterfactual communication setups which eliminates the weak trace, are discussed.
A recently proposed quantum protocol for counterfactual communication [Y. Aharonov and L. Vaidman, Phys. Rev. A 99, 010103(R) (2019)] relies on postselection to eliminate the weak trace in the transmission channel. We show that the postselection in this protocol additionally eliminates the flow of Fisher information from transmitter to receiver. However, we also show that a classical communication protocol with postselection can be counterfactual. Hence, we argue that postselection should not be allowed in genuine quantum counterfactual communication. In the proposed quantum protocol, the probability of discarding an event by postselection tends to zero with an increasing number of ideal optical components. But the counterfactual violation strength tends to infinity at a faster rate. Consequently, the quantum protocol is not counterfactual proper.
We provide an in-depth investigation of parameter estimation in nested Mach-Zehnder interferometers (NMZIs) using two information measures: the Fisher information and the Shannon mutual information. Protocols for counterfactual communication have, so far, been based on two different definitions of counterfactuality. In particular, some schemes have been based on NMZI devices, and have recently been subject to criticism. We provide a methodology for evaluating the counterfactuality of these protocols, based on an information-theoretical framework. More specifically, we make the assumption that any realistic quantum channel in MZI structures will have some weak uncontrolled interaction. We then use the Fisher information of this interaction to measure counterfactual violations. The measure is used to evaluate the suggested counterfactual communication protocol of H. Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)]. The protocol of D. R. M. Arvidsson-Shukur and C. H. W. Barnes [Phys. Rev. A 94, 062303 (2016)], based on a different definition, is evaluated with a probability measure. Our results show that the definition of Arvidsson-Shukur and Barnes is satisfied by their scheme, while that of Salih et al. is only satisfied by perfect quantum channels. For realistic devices the latter protocol does not achieve its objective.
The 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, has been known for quarter of a century. The protocols of full communication, including transmitting unknown quantum states, were proposed only a 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 being 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.
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 develop a quantum protocol that allows Alice to perceive Bob's message ``counterfactually''; that is, without Alice receiving any particles that have interacted with Bob. By utilizing a setup built on results from interaction-free measurements, we outline a communication protocol whereby the information travels in the opposite direction of the emitted particles. In comparison to previous attempts on such protocols, this one is such that a weak measurement at the message source would not leave a weak trace that could be detected by Alice's receiver. While some interaction-free schemes require a large number of carefully aligned beam splitters, our protocol is realizable with two or more beam splitters. We demonstrate this protocol by numerically solving the time-dependent Schr\"odinger equation for a Hamiltonian that implements this quantum counterfactual phenomenon.
Counterfactual communication, i.e., a communication without particles traveling in the transmission channel, is a bizarre quantum effect. Starting from interaction-free measurements many protocols achieving various tasks from counterfactual cryptography to counterfactual transfer of quantum states were proposed and implemented in experiments. However, the meaning of conterfactuality in various protocols remains a controversial topic. A simple error-free counterfactual protocol is proposed. This protocol and its modification are used as a test bed for analysis of the meaning of counterfactuality to clarify the counterfactuality status of various counterfactual proposals.
The counterfactuality of the recently proposed protocols for direct quantum communication is analyzed. It is argued that the protocols can be counterfactual only for one value of the transmitted bit. The protocols achieve a reduced probability of detection of the particle in the transmission channel by increasing the number of paths in the channel. However, this probability is not lower than the probability of detecting a particle actually passing through such a multi-path channel, which was found to be surprisingly small. The relation between security and counterfactuality of the protocols is discussed. An analysis of counterfactuality of the protocols in the framework of the Bohmian interpretation is performed.