To study the performance of this CFC protocol we measure the average bit error, as a function of the number of photons in which the bit is encoded, M, for five different values of N number of BSs. For the logic 0, we configure the MZIs in Bob’s laboratory as mirrors (see Fig. 2), while for the logic 1 we let the MZIs in Bob’s laboratory act as SWAP gates, routing the light out of the interferometer chain. Since Alice cannot access detector D B , she assumes that a photon is injected in the transmission channel every time she detects a heralding photon in D H . We thus run the measurement until we have M recorded single-photon events in D H (typical rates were 1.1 MHz) and look for the coincidences that these events have with D A within a set coincidence window Δτ = 2.5 ns that is shorter than the pulse separation. Our heralding efficiency was ~3% through the PNP.

Figure 3a shows the experimental average error probability of our CFC protocol as a function of M for different N. We also include a theoretical calculation of the expected error probabilities, which considers the heralding efficiency of the single photons and the success probability of the interferometer that is in good agreement with the experimental data. Note that these are not fits to the data, but rather models with no free parameters. As theoretically predicted, the error rate of the logic 1 decreases exponentially with increasing M and the error rate of the logic 0 increases linearly with M. We observe that higher N requires smaller M, and also results in lower bit error probabilities.

Fig. 3 Success probabilities of the CFC communication. The curves are theoretical models of our experiment with no free parameters, and the points are experimental data. a Measured average bit error (as defined in the main text) of the protocol for different number of beamsplitters (N) as a function of the number of photons (M) used to encode each bit. For small M the cos2N(π/2N) dependence of the logic 1 error dominates the average error, making the latter decrease with M as expected. As M is increased more, the linearly growing error in the logic 0, caused by imperfect destructive interference in Alice’s port (D A ), starts to dominate. b In the N = 6 case, the optimization of the interferometer fidelity and heralding efficiency leads to an average bit error rate of 1.5% for M = 320, where the average CFC violation probability is 2.4% Full size image

The success probability of this CFC scheme is highly sensitive to the fidelity of the interferometers and the overall heralding efficiency, which depends on the single-photon source and the coupling efficiency throughout the system. Hence, we optimized the setup for the N = 6 case. Figure 3b shows the corresponding error probability of the logic 1 and the logic 0. The inset in Fig. 3b shows the average error probability, where we find a minimum of 1.5% for M = 320, while the average counterfactual violation is kept at 2.4%. Owing to backscattering in Bob’s laboratory (i.e., imperfect SWAP operations) small “amounts” of wavefunction amplitude leak back into the transmission line in the 1 bit process. Although these do not all lead to detection events in Alice’s laboratory, the sum of their squares provides an upper bound on the probability of a counterfactual violation. We estimate that the probability for a photon to reflect off of a SWAP operation is at most 1%. Hence, in our experiment (Fig. 4) with M = 320 and N = 6, the weak trace is vanishingly small and the contribution from the logic 1 to a CFC violation is less than 1.1%. Note that this violation probability decreases with N, even if the errors remain the same.

Fig. 4 Image sent from Bob to Alice. The bits are encoded in different numbers of single photons M = {10, 50, 320, 500}. The white and black pixels are defined to correspond to logic 1 and logic 0, respectively. The success probability increases with increasing M, reaching 99% for M = 320. The CFC violation probability (P CFC ) also increases with increasing M, but it remains as low as 0.6% for M = 320. Note that this CFC violation comes only from the logic 0 errors, which we can directly measure; the total CFC violation would include a small portion of successful logic 1 events, as discussed in the main text. Increasing M beyond 320 increases the success probability at the expense of increasing the CFC violation. As it can be observed, these probabilities are directly related to the transmission fidelity (F) of the white pixels, which increases with M, and the transmission fidelity of the black pixels, which decreases with M Full size image

To demonstrate the performance of the communication protocol we proceed to analyze the quality of a message in the form of a black and white image, sent from Bob to Alice, for N = 6 and M = {10, 50, 320, 500}. We arbitrarily define the white and black pixels of the image as logic 1 and logic 0, respectively.

Figure 4 shows the message transmitted from Bob to Alice for different numbers of encoding photons. We define the image fidelity as

$$F = \mathop {\sum}\limits_{i = 1}^T {\frac{{1 + ( - 1)^{A_i + B_i}}}{{2T}}}$$ (3)

where B i is the bit that Bob intended to send, A i is the bit that Alice recorded, and T is the total number of bits in the image. In this case we define the CFC violation probability as the number of incorrectly transmitted logic 0 s (black pixels) over T. The encoding using M = 10 is clearly not enough to overcome the losses of the system, with a very low image fidelity of 31.77%. As we increase M, the success probability and legibility of the message increases (the individual fidelities are listed below each panel). The image fidelity reaches 99.09% at M = 320, at which point the CFC violation probability from 0 bit errors remains as low as 0.6%. For M = 500 the image fidelity does not noticeably change; however, the CFC violation increases slightly. If the CFC violation of the 1 bit (caused by on-chip beamsplitter imperfections) is accounted for, the CFC violation at M = 320 increases to 2.3%. Note that these values are lower than the value in Fig. 3b due to the unbalanced distribution of black and white pixels in the image.

Our high-fidelity implementation of a counterfactual communication protocol without post-selection was enabled by a programmable nano-photonic processor. The high (99.94%) average visibility of the individual integrated interferometers allowed bit error probabilities as low as 1.5%, while, at the same time, keeping the probability for the transmission of a single bit to result in a counterfactual violation below 2.4%. By combining our state-of-the-art photonic technology with a novel theoretical proposal we contradicted a crucial premise of communication theory:26 that a message is carried by physical particles or waves. In fact, our work shows that “interaction-free non-locality”, first described by Elitzur and Vaidman,2 can be utilized to send information that is not necessarily bound to the trajectory of a wavefunction or to a physical particle. In addition to enabling further high-fidelity demonstrations of counterfactual protocols, our work highlights the important role that technological advancements can play in experimental investigations of fundamentals of quantum mechanics and information theory. We thus anticipate nanophotonic processors, such as ours, to be central to future photonic quantum information experiments all the way from the foundational level to commercialized products.