Post-Selection and Counterfactual Communication

TL;DR

This paper critically examines quantum counterfactual communication protocols that rely on post-selection, arguing that such protocols are not genuinely counterfactual because post-selection eliminates information flow and the violation strength diverges.

Contribution

The paper demonstrates that post-selection removes Fisher information flow in quantum protocols and shows classical post-selected protocols can be counterfactual, questioning the genuineness of quantum counterfactual communication.

Findings

Post-selection eliminates Fisher information flow.

Classical post-selected protocols can be counterfactual.

Quantum protocols' violation strength diverges faster than discard probability.

Abstract

A recent quantum protocol for counterfactual communication [Y. Aharonov and L. Vaidman, Phys. Rev. A 99, 010103(R), 2019] relies on post-selection to eliminate the weak trace in the transmission channel. We show that the post-selection in this protocol also eliminates the flow of Fisher information from transmitter to receiver. However, we also show that a classical communication protocol with post-selection can be counterfactual. Hence, we argue that post-selection should not be allowed in genuine counterfactual communication. In the quantum counterfactual communication protocol, the probability of discarding an event by post-selection 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.

Figures

Equations

$$F(\theta )=\sum _i{p}_i(\theta )[{\partial }_{\theta }\ln p_i(\theta ){]}^2,$$

$${\mathcal{D}}_{vio}=n_{\gamma }\underset{\overline{\theta }\to 0}{\lim }\sum _{t=1}^{N_T}\frac{F^0({\theta }_t)+F^1({\theta }_t)}{2F_{ref}},$$

$${\mathcal{D}}_{vio}^0=$$

$$\displaystyle\times\sum_{m=1}^{M-1}\sin^{2}\Big{(}\frac{m\pi}{2M}\Big{)}.$$

$${\mathcal{D}}_{vio}^0\approx$$

Full text

††preprint: APS/123-QED

I Introduction

In standard communication protocols, a message is carried by signal particles propagating from the information transmitter to the information receiver Shannon (1948). Recent years have seen an intense debate about whether or not quantum mechanics can allow a message to be sent without the transmitter sending any particles to the receiver—that is, counterfactually Salih et al. (2013); Gisin (2013); Vaidman (2014); Salih et al. (2014); Li et al. (2015); Vaidman (2015); Arvidsson-Shukur and Barnes (2016); Vaidman (2016); Li et al. (2016); Griffiths (2016); Arvidsson-Shukur et al. (2017a); Calafell et al. (2018).

A protocol, first proposed by Salih et al., is based on a nested concatenated Mach-Zehnder interferometer (MZI) and the quantum Zeno effect Salih et al. (2013). Superficially this protocol achieves so-called type-@slowromancapi@ counterfactual communication (CFC): the quantum celebrities Alice (the information receiver) and Bob (the information transmitter) can communicate without any particles crossing the transmission channel that separates them. This protocol has been subject to intense criticism. First, the protocol requires an interferometer with tens of thousands of optical components to achieve high ($>95\,\%$) success probabilities Arvidsson-Shukur and Barnes (2016). Second, an analysis of the flow of Fisher information shows that counterfactuality is satisfied only in the presence of perfectly noiseless quantum channels Arvidsson-Shukur et al. (2017a). Third, Alice’s particles leave a weak trace in Bob’s laboratory, which according to several scholars means that the particles crossed the transmission channel Vaidman (2013); Danan et al. (2013); Vaidman (2014); Aharonov et al. (2017). Fourth, unless an infinite number of perfect optical components is used, post-selection has to be employed in order to discard events that violate counterfactuality Calafell et al. (2018).

In their innovative article, Ref. Aharonov and Vaidman (2019), Aharonov and Vaidman (AV) modify the counterfactual communication protocol of Salih et al. Salih et al. (2013), claiming to make it robust against the weak-trace argument. Their interferometer protocol has two processes, by which Bob can send a [math]-bit or a $1$-bit to Alice, with some probability of success. If such a process succeeds, the particle that Alice detects left no weak trace Vaidman (2013) in Bob’s part of the interferometer, meaning that Bob can not infer the particle’s presence via a local weak measurement.

In this paper we use methods from information theory to study the protocol of AV. In particular we study the flow of Fisher information in the interferometers used by AV. We find that in successful bit-transmission processes, the protocol particles carry no Fisher information about Bob’s laboratory to Alice. From a Fisher information perspective, the AV protocol is counterfactual if post-selected on successful bit-transmissions. However, we argue that this is not enough to comply with the original type-@slowromancapi@ definition of CFC, which does not consider any post-selection requirement Salih et al. (2013). We design a classical protocol, which can achieve CFC with post-selection, and conclude that post-selection is undesirable if CFC is to describe a non-classical phenomenon. We then show that the events, which are discarded by post-selection in the AV protocol, violate counterfactuality to a level where the protocol cannot be called counterfactual proper.

II Fisher-information measure of counterfactuality

To evaluate the level of counterfactuality of an experiment we need a measure of inter-measurement particle presence. One conceptually beautiful suggestion of such a measure is the weak trace Vaidman (2013). However, in the words of Vaidman, et al.: ‘The weak value requires the two-state vector formalism for its definition’ Vaidman et al. (2017); and the two-state vector formalism is a controversial interpretation of quantum mechanics, which has generated criticism of the weak-trace measure Svensson (2013); Li et al. (2013, 2016); Hashmi et al. (2016).

In Ref. Arvidsson-Shukur et al. (2017a) we (together with A. Gottfries) suggested how the flow of Fisher information can be used to study the inter-measurement presence of a quantum particle, from an interpretation-independent and operational perspective. The Fisher information about an unknown parameter, $\theta$, is

[TABLE]

where $p_{i}(\theta)$ is the probability of measurement outcome $i$ given $\theta$ Cover and Thomas (2006).

Consider a horizontally polarised photon travelling through a non-polarising interferometer. Perfectly noiseless quantum channels are purely fictional, and we introduce a vanishingly weak polarisation rotation, a “tagging”, to mimic an unwanted polarisation disturbance Arvidsson-Shukur et al. (2017b). The tagging is the only component acting on the polarisation degree of freedom in the interferometer. We thus argue that output photons, which carry polarisation-encoded information about the tagging, have been present at the location of the tagging. This toy model can be used to study how information flows through interferometers. We quantify the extent to which a particle has been present at the location of the tagging by $\mathcal{D}_{vio}=F(\theta)/F_{ref}$, where $\theta$ is the parameter that sets the tagging interaction Arvidsson-Shukur et al. (2017a). $F_{ref}$ is the reference Fisher information obtained had the whole particle wavepacket propagated freely through the location of the interaction. $\mathcal{D}_{vio}\geq 1$ reveals a full counterfactual violation.

Here we analyse interferometers where the particle can move in and out of Bob’s laboratory several times. Every time the particle does so, type-@slowromancapi@ counterfactuality is violated. We extend the Fisher-information measure of the counterfactual violation strength from Ref. Arvidsson-Shukur et al. (2017a) to the scenario of multiple paths through Bob’s laboratory:

[TABLE]

where $n_{\gamma}$ is the average number of interferometer evaluations per transmitted bit, and $\overline{\theta}\equiv(\theta_{1},\,...,\,\theta_{N_{T}})$ is a vector containing all the parameters that set the $N_{T}$ tagging interactions placed at each point where the protocol particle enters or re-enters Bob’s laboratory. $F^{0}$ and $F^{1}$ are the Fisher information in a [math]-bit and a $1$-bit process, respectively, calculated from the output probabilities of the interferometer. We assume messages with a balanced number of [math]- and $1$-bits. In experiments without post-selection, our measure is directly proportional to the average sum of the integrated probability density that travels in and out of Bob’s laboratory per transmitted bit (in the Schrödinger picture).

III A classical CFC protocol with post-selection

In type-@slowromancapi@ CFC, a message is sent without any particles crossing the transmission channel between Alice and Bob Salih et al. (2013); Arvidsson-Shukur et al. (2017a). Below we show that a classical protocol can transmit two bit values counterfactually if post-selection is allowed.

Alice and Bob reside in respective laboratories, connected with a pipe, through which Bob can roll balls to Alice. The communication protocol is as follows

• •

The message is sent by Bob rolling balls to Alice. One ball, and bit, is transmitted per minute. $T_{r}<1\,\textrm{min}$, where $T_{r}$ is the time it takes one ball to roll from Bob to Alice.

• •

On even minutes, if Bob wishes to transmit a $1$-bit ([math]-bit) he rolls (does not roll) a ball through the pipe to Alice.

• •

On odd minutes, if Bob wishes to transmit a [math]-bit ($1$-bit) he rolls (does not roll) a ball through the pipe to Alice.

At even (odd) minutes, Alice records the value of the [math]-bit ($1$-bit) without receiving Bob’s protocol particles (balls), and without these particles crossing the transmission channel (the pipe). These bit values satisfy the type-@slowromancapi@ definition of CFC. Post-selecting on Alice not receiving any particles, she receives all message bits counterfactually. We thus conclude that CFC with post-selection is not an inherent quantum phenomenon. A definition of non-classical CFC should require that the counterfactuality stems from quantum phenomena, rather than classical post-selection.

IV Reduced AV CFC Protocol

Vaidman and Aharonov first investigate a reduced CFC protocol, based on a doubly nested MZI. See Fig. 1. The outer and inner beam-splitters have transmission probabilities of $4/5$ and $1/2$, respectively. Bob can insert mirrors or detectors in the upper path of the interferometer. The inner MZIs are tuned to give completely destructive interference towards their lower output paths when the mirrors are inserted. Both bit-transmission processes start by Alice sending a photon into the interferometer’s lower path. To send a [math]-bit, Bob inserts his mirrors. To send a $1$-bit, Bob replaces his mirrors with detectors. Only the [math]-bit process can probabilistically trigger detector $D_{0}$. The tagging interactions rotate the polarisation state: $\ket{H}\rightarrow\cos(\theta)\ket{H}+\sin(\theta)\ket{V}$ and $\ket{V}\rightarrow\cos(\theta)\ket{V}-\sin(\theta)\ket{H}$, with $\theta\ll 1$. The reference Fisher information is $F_{ref}=4$.

To satisfy counterfactuality, the reduced AV protocol requires the post-selection of only the events that trigger detectors $D_{0}$ or $D_{1}$. Calculating the Fisher information from the post-selected outcome probabilities, we find that $F(\theta_{1})$ and $F(\theta_{2})$ are strictly zero in the $1$-bit process and approaches zero as $\theta_{1},\,\theta_{2}\rightarrow 0$ in the [math]-bit process. From a post-selected perspective, the counterfactual violation strength is indeed approaching zero for both bit values. However, as we argued above, post-selection is undesirable in CFC.

We thus proceed by calculating $F(\theta_{1})$ and $F(\theta_{2})$ with respect to the measurement statistics of all the detectors of the device, without post-selection. We find that $F^{0}(\theta_{1})=\frac{8}{5}$ and $F^{0}(\theta_{2})=\frac{4}{5}[1-\cos(\theta_{1})]$ in the [math]-bit process, and $F^{1}(\theta_{1})=\frac{8}{5}$ and $F^{1}(\theta_{2})=\frac{2}{5}$ in the $1$-bit process. The value of the type-@slowromancapi@ counterfactual violation strength is then $\mathcal{D}_{vio}=n_{\gamma}\frac{9}{20}$.

Finally, let us estimate $n_{\gamma}$. In Ref. Aharonov and Vaidman (2019), the bit values are deduced from the statistics of detector $D_{0}$ only. For a successful [math]-bit transmission, $D_{0}$ must be triggered. In one [math]-bit process the probability for this is $P_{D_{0}}=1/25$. For the average [math]-bit error to be less than $5\,\%$, we repeat each process $n_{\gamma}>74$ times. This gives a counterfactual violation strength of $\mathcal{D}_{vio}>33.3$; the non-post-selected reduced protocol is not counterfactual.

V Full AV CFC Protocol

By increasing the number of perfect optical components it is possible to improve the probability of successful bit-transmission in the AV CFC protocol Salih et al. (2013); Aharonov and Vaidman (2019). The extended interferometer is similar to that in Fig. 1, but concatenates $N$ beam-splitters in the outer MZI, and concatenates $M$ beam-splitters in each of the inner MZIs Aharonov and Vaidman (2019). The outer and inner beam-splitters have transmission probabilities $\sin^{2}(\pi/2N)$ and $\sin^{2}(\pi/2M)$, respectively. To transmit bits with high success probabilities the protocol requires that $M\gg N\gg 1$. For perfect optical components, and if $M,\;N\rightarrow\infty$, the success probability of both bit processes approaches unity, and $n_{\gamma}=1$. This is an unrealistic protocol, and not “a very reliable communication protocol” as suggested in Ref. Aharonov and Vaidman (2019).

We consider vanishingly weak tagging interactions at each particle path entering or re-entering Bob’s part of the interferometer. As in the reduced AV protocol, the Fisher information, and thus also counterfactual violation strength, approaches zero when we post-select on outcomes at $D_{0}$ and $D_{1}$. But, does the counterfactuality of the AV CFC protocol stem from a genuine quantum phenomenon, or simply from the post-selection of specific outcomes?

Let us analyse the full AV CFC protocol without post-selection. For the $1$-bit process, the counterfactual violation is equal to the probability of detection in Bob’s laboratory, which approaches zero for large numbers of beam-splitters. For the [math]-bit process the counterfactual violation is

[TABLE]

For large values of $N$ and $M$ this simplifies significantly:

[TABLE]

such that $\mathcal{D}_{vio}^{0}\gg 1$ for $M\gg N\gg 1$. We recall that if $\mathcal{D}_{vio}>1$ the protocol is not counterfactual. Consequently, the AV CFC protocol is invalid without post-selection, even if $M,\;N\rightarrow\infty$. An analysis of the weak trace in the events discarded by post-selection leads to a similar conclusion. We can thus conclude that the quantum interferometer used in the AV CFC protocol does not eliminate counterfactual violations. This elimination is acquired instead by deploying post-selection—as in the classical CFC protocol above.

VI Discussion

Aharonov and Vaidman write: “The possibility to communicate between spatially separated regions, without even a single photon passing between the two parties, is an amazing quantum phenomenon” Aharonov and Vaidman (2019). Here, we have shown that CFC with post-selection can be achieved classically too, without any “quantum phenomenon”. Hence, we argue that a truly counterfactual quantum experiment should operate without post-selection. The AV CFC protocol fails this requirement.111The same is true for the post-selected CFC protocol suggested in Ref. Salih et al. (2018).

The AV protocol is nevertheless interesting. The post-selection in this protocol is different than in our classical protocol. In the classical protocol the probability of discarding a non-counterfactual event is $1/2$. In the AV protocol, the rate of discarding message bits by post-selection decreases with an increasing number of optical components. The surprising result is that the counterfactual violations can be distilled to a small number of events.222See Ref. Arvidsson-Shukur et al. (2019) for an outline of how post-selection can distill Fisher information beyond classical limits. However, with the same increase of optical components, the counterfactual violation strength of these events tends to infinity at a faster rate making the AV protocol violate counterfactuality if post-section is prohibited.

Finally, we would like to direct the reader’s attention to our own definition of CFC: the type-@slowromancapii@ definition Arvidsson-Shukur and Barnes (2016); Arvidsson-Shukur et al. (2017a). This definition also considers a single-particle protocol, but relaxes slightly the restrictions on the particle propagation. Particles are allowed to travel in the opposite direction to the message, from information receiver to information transmitter. But particles are not allowed to travel from the transmitter to the receiver. In our type-@slowromancapii@ CFC protocol information and particles counterpropagate. A recent work, Ref. Calafell et al. (2018), experimentally demonstrate this non-classical communication protocol, without post-selection, and with a success rate of $\sim 99\,\%$.

The authors would like to thank Irati Alonso Calafell for useful discussions. D.R.M.A.-S. acknowledges support from the EPSRC, the Sweden-America Foundation, and the Lars Hierta Memorial Foundation.