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arXiv:2301.01730 · Physical Review A

Multitime quantum communication: Interesting but not counterfactual

Robert B. Griffiths

Published Jun 28, 2023• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.107.062219

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In a recent paper, Griffiths [Phys. Rev. A 107, 062219 (2023)] analyzed a protocol for transmission of information between two parties introduced by Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)]. There is a considerable controversy about the counterfactuality of this protocol, and Griffiths suggested to resolve it by introducing a new measure of channel usage, which he called ``Cost.'' I argue that this measure is not appropriate because the original interaction-free measurement protocol which triggered the definition of the concept of counterfactuality is not counterfactual according to this measure.

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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.
In a recent paper, Griffiths [Phys. Rev. A 107, 062219 (2023)] analyzed a protocol for transmission of information between two parties introduced by Salih et al. [Phys. Rev. Lett. 110, 170502 (2013)]. There is a considerable controversy about the counterfactuality of this protocol, and Griffiths suggested to resolve it by introducing a new measure of channel usage, which he called ``Cost.'' I argue that this measure is not appropriate because the original interaction-free measurement protocol which triggered the definition of the concept of counterfactuality is not counterfactual according to this measure.
The field of quantum mechanics has revolutionised physics as a subject. Areas such as information theory, computer science and physical sensing have all been affected by the tremendous successes of various quantum protocols. In this thesis I present my contribution to the development of such non-classical protocols. In classical communication theory a message is always carried by physical particles that interact with a transmitter, after which they travel to a receiver. In this thesis I outline a quantum protocol which allows a receiver to obtain a message without receiving any physical object or particles that have interacted with the transmitter—that is, counterfactually. I build my protocol for counterfactual communication on the principles of interaction-free measurements, ensuring that information always propagates in the opposite direction to the protocol particles. The protocol shows how quantum mechanics breaks the previous premise of communication theory. From the perspective of local observers, it is a beautiful manifestation of the non-locality of interaction-free measurements. Furthermore, it is highly robust against experimental errors and external disturbances. The majority of this part of the thesis is based on my published article ‘Quantum counterfactual communication without a weak trace’ [Phys. Rev. A 94, 062303 (2016)]. Previous to my work, Salih et al. attempted to design a counterfactual communication protocol [Phys. Rev. Lett. 110, 170502 (2013)]. This protocol has been highly controversial. As counterfactual phenomena impose restrictions on the inter-measurement paths of quantum particles, and the physical reality of such paths lacks description in the Copenhagen interpretation of quantum mechanics, an extension of current quantum theory is required to facilitate a discussion. In this thesis I present an operational and interpretation-independent methodology, enabling the discussion of inter-measurement paths of quantum particles. I start by considering the interferometers of counterfactual protocols, making the basic assumption that any quantum evolution naturally involves uncontrolled weak interactions. I then show how the Fisher information of these weak interactions, available at the output of counterfactual experiments, can be used to discuss the pre-measurement past of the particles. Based on this analysis, the protocol developed by Salih et al. is found to strongly violate counterfactuality. However, my protocol is more flexible in that it allows particles to propagate in the opposite direction to the message. This leads to counterfactuality being satisfied—even in the presence of large experimental errors. These results are observed both analytically and numerically. This part of the thesis is based on my article ‘Evaluation of counterfactuality in counterfactual communication protocols’ [Phys. Rev. A 96, 062316 (2017)]. The numerical methods are inspired by another of my publications: ‘Protocol for fermionic positive-operator-valued measures’ [Phys. Rev. A 96, 052305 (2017)]. Finally, as the Fisher information measure is found to be useful in evaluating counterfactual protocols, I extend my work by investigating the quantum Fisher information in experiments with general discrete quantum circuits. I prove that the quantum Fisher information of a two-level interaction in a quantum circuit can be expressed by a simple formula. Under certain phase-relations, the formula provides a straightforward connection between the abstract concept of the inter-measurement wavefunction and the quantum Fisher information at the output. With regard to how the information obtained from a certain volume of space influences our perception of classical objects, I argue that the quantum Fisher information measure is highly useful in describing quantum objects. If this measure is applied to observers with a limited set of the experimental measurement outcomes, a quantum object can appear to follow non-classical discontinuous paths. This supports the remarkable conclusion that our perception of the past of a quantum object is subjectively dependent on the measurement we conduct on it.
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.
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.
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.
Abstract Quantum Counterfactual Communication is the recently-proposed idea of using quantum physics to send messages between two parties, without any matter/energy transfer associated with the bits sent. While this has excited massive interest, both for potential ‘unhackable’ communication, and insight into the foundations of quantum mechanics, it has been asked whether this process is essentially quantum, or could be performed classically. We examine counterfactual communication, both classical and quantum, and show that the protocols proposed so far for sending signals that don’t involve matter/energy transfer associated with the bits sent must be quantum, insofar as they require wave-particle duality.