Counterfactual communication without a trace in the transmission channel

These counterfactual communication protocols crucially rely upon interaction-free detection (IFD), which was originally introduced in a paper ⁶ co-authored by Vaidman, who is also an author of the present paper. However, Vaidman has repeatedly noted that IFD can only verify the existence of an object at a location, not its absence. This corresponds to a one-valued counterfactual transfer which is enough for counterfactual key distribution ^{7, 8}, but does not allow general counterfactual computation ¹, communication of classical data ², or quantum state transfer ^{3, 4}. For these proposals and their experimental implementation ⁹, Vaidman showed ^{10, 11, 12} that when the absence of an object in a particular place is established, the particle leaves a trace en route similar to the trace of a single localized particle.

It is simple to show why it impossible to devise a protocol that unambiguously distinguishes between blocking and not blocking one particular location $A$ without leaving a first-order trace in that location. Alice sends a photon that can be detected in detector $D_{0}$ or $D_{1}$. Placing an object in the location $A$ must make detection in $D_{0}$ impossible. By our rules, the object can only absorb and not reflect, so this change tells us that there is a path through $A$ from Alice’s source to the detector $D_{0}$. Therefore, there have to be a path from the source to $A$ and also a path from $D_{0}$ to $A$. In such a situation, the two-state vector formalism (TSVF) ¹³, which adds to the description of the photon a quantum state evolving backward in time from the postselection at the detector, provides a simple way to find the location of the weak traces left by the photon. The overlap of the forward and backward evolving states of the photon at $A$ leads to the trace in $A$ of the first order in the interaction parameter ^{14, 15}, similar to the trace of a photon localized at $A$. This makes it unreasonable to claim that the photon was not there, i.e. that this was a counterfactual communication.

The experiment in this paper addresses this critical weakness of earlier protocols. We implement a protocol design based on the modification of these protocols found recently by Aharonov and Vaidman (AV) ¹⁶ that eliminates the trace of the first order also in the case of absence of the object, therefore permitting communication that is counterfactual according to our more stringent weak trace criterion.

The key ingredient of the AV modification of counterfactual protocols is that Bob communicates his bit to Alice by blocking or not blocking two locations $A_{1}$ and $A_{2}$ together, instead of one location $A$, see Fig. 1. Alice gets a click in detector $D_{0}$ when $A_{1}$ and $A_{2}$ are not blocked, but it must be impossible when locations $A_{1}$ and $A_{2}$ are blocked. Therefore, there must be paths from the source to the detector through $A_{1}$ and $A_{2}$ that cause destructive interference of the waves from all the paths together. There are three such paths: through $A_{1}$ and $A_{2}$, through $A_{1}$ but not $A_{2}$, and through $A_{2}$ but not $A_{1}$. These additional paths provide a finite contribution of the amplitude of the photon wave in the detector $D_{0}$. However, the waves in these three paths interfere destructively in both $A_{1}$ and $A_{2}$. There is no overlap of the forward and backward evolving wave functions at $A_{1}$ and $A_{2}$, and consequently, no first order trace is created. Note that blocking only one of the locations $A_{1}$ or $A_{2}$, does not lead to a possibility of detection at $D_{0}$, so the argument of non-counterfactuality of the non-modified protocols does not hold here.

In our experiment, we do not only perform the counterfactual communication protocol with AV modification, but we also verify that both logical bit values are communicated without leaving a significant trace in the connecting path of the transmission channel. By applying small and distinct frequency shifts to each photon at different locations and filtering by frequency during collection ^{17, 18}, we show where those photons interacted significantly to leave a trace and where they did not. We verify that any trace left by information-carrying photons in the transmission channel is small enough to not be visible above the noise floor, unlike the much larger traces left by single photons during calibration.

Our experimental results are a proof of principle. The transferred bits have high enough fidelity to send a QR code image that can be scanned by a cell phone, but the current design has a small success probability for the sent photons to be detected. Since the stray photons may be intercepted, this simple design is not yet secure; however, the application of our modification to counterfactual protocols based on the quantum Zeno effect ⁹ could dramatically increase the success probability, potentially enabling secure communication.

To explain the basic mechanism for achieving counterfactual communication, we first consider the setup ¹⁶ shown in Fig. 1. It has a nested pair of Mach-Zehnder interferometers (MZIs) that will be conceptually folded onto each other to obtain the compact Michelson-Morely interferometer (MMI) shown in Fig. 2, which our experiment implements with heralded single photons and active stabilization as shown in Fig. 3.

The left path in Fig. 1 is one arm of an (outer) MZI, so each photon emitted by the source at the top left bounces off three mirrors before reaching the final beamsplitter at the bottom, with detector $D_{0}$ placed at the right output port. We then split the right arm of this outer MZI into two consecutive (inner) MZIs, with detector $D_{1}$ placed at the right output port of the second inner MZI. As shown in Fig. 1(a), we tune each inner MZI to destructively interfere at the double-sided mirror that connects them, allowing all photons in the right arm to escape before they can trigger detector $D_{1}$. Thus, each photon must either escape or cause the detector $D_{0}$ to click. However, if we block the right arms of the inner MZIs with shutters, as shown in Fig. 1(b), the remaining unblocked path will either trigger detector $D_{1}$ or complete the outer MZI, which we tune to destructively interfere at its right output and thus prevent the detector $D_{0}$ from clicking.

The unfolded design in Fig. 1 also illustrates both forward- (red) and backward-evolving (green) wave function paths, such that their overlap predicts which traces may be seen in the experiment. In regions where the two overlap (gray), a weakly interacting probe will record a trace consistent with those observed for definite particle paths. However, the same probe will not register an appreciable trace where there is no overlap. Notably, this simple picture predicts that no trace will be observed in the right arms of the inner MZIs, even though they form the transmission channel for information about the block.

For simplicity of implementation, we conceptually fold the design in Fig. 1 over a horizontal line on the double-sided mirror to identify both inner MZI and obtain Fig. 2. This fold converts the outer MZI into an MMI so that each photon passes through the same inner MZI twice. The benefit of this compact variation is that only one shutter is needed to change the bit configuration, since each photon may reach that shutter from two different directions. The penalty is that the transmissivity of the beamsplitters must also be slightly adjusted when the shutter is inserted to preserve the visibility of the destructive interference. For simplicity, we use 50/50 beamsplitters and balance the intensities with a tunable attenuator placed in the lower arm ($C$).

To interrogate each photon about where it has been without spoiling the sensitive interference, we arrange for the photon to acquire small and distinct frequency-shifts when traversing different arms of the interferometer. We then spectrally filter those photons before detection to obtain frequency-dependent intensities that reveal their past interaction history. Following Zhou et al.¹⁷ we use free-space frequency electro-optic modulators (EOM) to induce frequency-shifted sidebands to mark the paths.

We provide additional technical detail about our implementation of Fig. 2 in the caption of Fig. 3 and the Methods section.

Since inserting a beam block changes which detector can click, this interferometric device enables the following one-way communication protocol. Let Alice only have access to the detectors $D_{0}$ and $D_{1}$, and Bob only have access to the right arms of the inner MZIs. Bob can communicate to Alice remotely using the interferometer if they agree beforehand on a common timing reference to label successive time bins. To account for the high probability of photon loss and the device imperfections, Alice and Bob pre-arrange time bin durations that will contain many redundant photon attempts per bin. For each agreed-upon time bin, Bob then sets the configuration for either bit 0 or bit 1 by removing or inserting his shutter, respectively. When photons are successfully detected by one of her detectors, Alice records both which detector clicked and the time bin of the click. The redundant attempts per bin permit Alice to successfully detect at least one trial for each bin, and thus receive Bob’s binary message. In principle, the communication fidelity can be made arbitrarily high for a given photon source by increasing the bin duration and thus the encoding redundancy.

It is instructive to analyze this situation by examining the forward- and backward-evolving wave functions shown in Fig. 1. For a successfully transmitted 0, i.e. $D_{0}$ clicks in Fig. 1(a), there is no overlap of the forward (red solid line) and backward (green solid line) wave functions in either the transmission channel or at Bob’s site. Similarly, for a successful 1, i.e. $D_{1}$ clicks in Fig. 1(b), there is still no overlap of the forward and backward wave functions in the transmission channel. Thus, we expect no measurable trace to be seen by weakly coupled probes placed in the transmission channel, at least to first-order in the small interaction strength ¹⁹.