Experimental quantum communication enhancement by superposing trajectories

In quantum communication networks, wires represent well-defined trajectories along which quantum systems are transmitted. In spite of this, trajectories can be used as a quantum control to govern the order of different noisy communication channels, and such a control has been shown to enable the transmission of information even when quantum communication protocols through well-defined trajectories fail. This result has motivated further investigations on the role of the superposition of trajectories in enhancing communication, which revealed that the use of quantum control of parallel communication channels, or of channels in series with quantum-controlled operations, can also lead to communication advantages. Building upon these findings, here we experimentally and numerically compare different ways in which two trajectories through a pair of noisy channels can be superposed. We observe that, within the framework of quantum interferometry, the use of channels in series with quantum-controlled operations generally yields the largest advantages. Our results contribute to clarify the nature of these advantages in experimental quantum-optical scenarios, and showcase the benefit of an extension of the quantum communication paradigm in which both the information exchanged and the trajectory of the information carriers are quantum.

Figure 1

Combining two channels in a superposition of trajectories. A sender and a receiver communicate under the restriction that the information carrier must cross at least one noisy region. (a) Quantum control of parallel channels. A quantum particle is placed in a quantum superposition of two trajectories, each branch containing a single noisy channel. (b) Channels in series with quantum-controlled operations. Each of the branches of the superposition passes through the noisy channels in the same order, but different unitary operations are applied locally in each branch. (c) Quantum control of channel order. The information carriers are routed through the two channels in different orders. This setup can achieve a genuinely indefinite order of the two channels. (d) Classical trajectories. Throughout this paper, we will compare the three quantum superpositions of channels above to classical trajectories. In this regard, if one has access to classical-like trajectories only, one can send the photon through one or the another noisy regions with probabilities $q$ and $1-q$.

Figure 2

Experimental setup. (a) Quantum control of parallel channels. After their polarization is set via a half wave plate (HWP) and a quarter wave plate (QWP), single photons are injected into a Mach-Zehnder interferometer. One noisy channel is placed into each arm of the interferometer, and each channel is realized through two liquid-crystal wave plates (LCWP), the first positioned at $0^{\circ }$ (to implement $\mathcal{I}$ or $Z$ by changing the retardance), the second at $22.5^{\circ }$ ($\mathcal{I}$ or $X$). By means of a piezoelectric trombone delay line, the photon interfering on the second beam splitter of the interferometer can be projected onto the bases ${\left\{\right|+\rangle }_{\mathrm{T}}{,|-\rangle }_{\mathrm{T}}\}$ or ${\left\{\right|R\rangle }_{\mathrm{T}}{,|L\rangle }_{\mathrm{T}}\}$ of the trajectory. Finally, the photons' polarization is measured through a sequence of QWP, HWP, and a polarizing beam splitter. (b) Channels in series with quantum-controlled operations. As in the previous scheme, the photons are prepared in polarization via QWP and HWP and injected into a Mach-Zehnder interferometer. In this case, the two noisy channels are placed in the two superposed branches in series with the same order. Also in this case, the channels are realized through LCWPs. Furthermore, before each noisy channel, additional unitary operations are realized through sequences of QWP, HWP, and QWP (before the first channel, the QWP, HWP, and QWP are placed in one branch of the trajectory only, whereas between the two channels the wave plates are in both branches, since we only implement cases where $U_2=U_3$). The rest of the setup is the same as in the previous case. (c) Quantum control of channel order. The preparation and measurement of the photons in polarization happens as in the previous schemes, as well as the realization of the noisy channels, and the projection of the trajectory DOF. In this case, however, the Mach-Zehnder interferometer is folded into two loops so that the photon can travel through the two channels in the two alternative orders in each arm of the interferometer. (d) Heralded single-photon source. We generate photon pairs using a type-II spontaneous-parametric downconversion source. One photon is directly detected with an avalanche photodiode (upper arm), whereas the other is coupled into an optical fiber and sent to one of the setups (a), (b), or (c). The interferometers in setups (a), (b), and (c) all contain two compensation HWP at the beginning and at the end of the reflected arm, so as to compensate for the phase shifts due to the reflection from the beam splitter.

Figure 3

Experimental $\mathit{XY}$ channel noise data. The theoretical trends associated with the channels in series with quantum-controlled operations and the quantum control of channel order show full activation. The experimental data do not perfectly match the theoretical trends because, for $p=0.5$, the channel produces an equal mixture of $X$ and $Y$ operations, and such case can be experimentally realized with a lower fidelity than the one in which only one of the two operations is performed (i.e., when $p=0$ or 1). It follows that, in the central region, the experimental data are further apart from the theoretical trend than they are on the upper end. The quantum control of parallel channels does not allow full activation, and thus it is positioned below the previous two trends. In this case, the experimental data are closer to the theoretical expectation. The reason of the higher agreement is that, in the case of the disposition of noisy channels in parallel, only one channel is present in each branch of the interferometer. As a consequence, the experimental imperfections affecting each branch are smaller than in the dispositions of channels in series and in indefinite order. Finally, the coherent information associated to only one $XY$ channel is theoretically lower than all the other layouts. A detailed analysis of the error estimation and the systematic error is provided in the Appendix, Sec. pp1-s5. The labels “QC-//-chann.,” “Series QC-op.,” and “QC-order” stand for “quantum control of parallel channels,” “channels in series with quantum-controlled operations,” and “quantum control of channel order,” respectively. The same labels will be used in all plots.

Figure 4

Experimental BF- and PF-noise data. The experimental data of quantum control of parallel channels and the quantum control of channel order are in good agreement with the theoretical trends. Conversely, the configuration of the channels in series with quantum-controlled operations shows a constant offset between the experimental data and the expected theoretical trend. This discrepancy is due to the fact that, in this case, all the liquid crystals are arranged in series, with the additional presence of wave plates realizing a Hadamard gate, and hence this configuration is the one that exhibits the greatest amount of experimental imperfections along each path. In spite of this, for most values of $p$ the coherent information that can be achieved with the series configuration is still above all others by several standard deviations.

Figure 5

Experimental BB84-channel noise data. As in the previous plots, the continuous lines show the expected theoretical trends, while the squares, circles, and crosses represent the experimental data corresponding to the quantum control of parallel channels, the channels in series with quantum-controlled operations, and the quantum control of channel order, respectively. All the experimental data are in high agreement with the expected theoretical trends.

Figure 6

Experimental characterization of a liquid-crystal wave plate (LCWP) at $0^{\circ }$. Since the crystal is positioned at $0^{\circ }$, it will be able to switch from an identity operation to a Pauli $Z$. To characterize the voltage corresponding to a Pauli $Z$, we send through it photons in the polarization basis $\left\{\right|\pm \rangle =\left(\right|0\rangle \pm |1\rangle )/\sqrt{2}\}$, and we measure for which voltage the population inversion occurs. The estimated errors are Poissonian.

Figure 7

Histogram of the coherent information achieved with the three channel layouts when the two copies of the same randomly generated channel is used in each of the three layouts. The histograms report the frequency with which a random channel ($y$ axis, in logarithmic scale) yields a given amount of coherent information ($x$ axis), normalized to the total number of channels used. (a) Histogram with ${10}^3$ bins between a coherent information of 0 and 0.85. As can be seen, the configuration of channels in series with quantum-controlled operations consistently achieves the highest coherent information on average. (b) Histogram of the same data with ${10}^5$ bins displayed for values of coherent information from 0 to 0.001. By increasing the resolution for small values of coherent information, it is possible to observe in greater detail the absence of the peak at zero for the quantum superposition of channels in series with quantum-controlled operations. In this region, the performance of the quantum control of parallel channels and that of quantum control of channel order is comparable.

Figure 8

Histogram of the coherent information achieved with the three channel layouts when two different randomly generated channels are used. The overall trend here is comparable to that of two copies of the same random channel (Fig. 7). However, in this case, the quantum control of the parallel channels performs, on average, better than the quantum control of channel order. Moreover, in general, all three layouts tend to perform worse than in the case of two copies of the same random noisy channel (i.e., the maximum amount of coherent information which can be achieved through each layout is generally lower than in the case shown in Fig. 7).

Figure 9

Histograms of the difference between coherent information achievable with quantum superposition of channels in series with quantum-controlled operations and the other two layouts in the case of (a) two independent copies of the same random channel, and (b) two different randomly generated channels. The histograms show, for each random channel, the difference between the coherent information of the quantum superposition of channels in series with quantum-controlled operations and that of the quantum control of parallel channels $[\left({\text{CI}}_{\text{Series}\text{QC-op.}}\right)-\left({\text{CI}}_{\text{QC-//-chann.}}\right)]$ and of quantum control of channel order $[\left({\text{CI}}_{\text{Series}\text{QC-op.}}\right)-\left({\text{CI}}_{\text{QC-order}}\right)]$. While, to a large extent, the layout using the channels in series with quantum-controlled operations tends to outperform the other two schemes, the negative values indicate that this is not always the case.

Figure 10

Monte Carlo simulation of the BB84 channel with $p=0.5$. A plot of the average process infidelity between the ideal process and the simulated process versus the number of applied operations used to simulate the noisy channel. The infidelity is defined as $1-F_{\text{av}}$, where $F_{\text{av}}$ is the fidelity. Hence, smaller infidelities indicate a higher degree of agreement.

Figure 11

Experimental BF- and PF-noise data for suboptimal quantum-controlled operations. The trend of the scheme featuring the channels in series with quantum-controlled operations (Series QC-op.) performs worse than the quantum control of channel order (QC-order) for all $p\ge 0.67$, but better than the quantum control of parallel channels (QC-//-chann.) for $p\ge 0.84$. The experimental data for the suboptimal choice of Series QC-op. is in good agreement with the expected trend.