Two-Way Covert Quantum Communication in the Microwave Regime

Quantum communication addresses the problem of exchanging information across macroscopic distances by employing encryption techniques based on quantum-mechanical laws. Here, we advance a new paradigm for secure quantum communication by combining backscattering concepts with covert communication in the microwave regime. Our protocol allows communication between Alice, who uses only discrete phase modulations, and Bob, who has access to cryogenic microwave technology. Using notions of quantum channel discrimination and quantum metrology, we find the ultimate bounds for the receiver performance, proving that quantum correlations can enhance the SNR by up to 6 dB. These bounds rule out any quantum illumination advantage when the source is strongly amplified, and shows that a relevant gain is possible only in the low photon-number regime. We show how the protocol can be used for covert communication, where the carrier signal is indistinguishable from the thermal noise in the environment. We complement our information-theoretic results with a feasible experimental proposal in a circuit-QED platform. This work makes a decisive step toward implementing secure quantum communication concepts in the previously uncharted $1$–$10$ GHz frequency range, in the scenario when the disposable power of one party is severely constrained.

Popular Summary

Quantum communication holds the promise of replacing the present classical communicating systems, but we are far yet from achieving a full quantum information network. Optical photons are the leading option for implementing these ideas, since unconditional security is provably reachable in this regime. On the other side, a microwave open-air quantum-communication application has been so far elusive. Here, we propose a novel implementation of quantum-secured backscatter communication in the microwave regime.

In backscatter communication, Alice modulates a carrier signal emitted by Bob to transmit its information without spending energy to generate the carrier. In order to hide the information transmitted by Alice from malicious Eve, it is necessary to disguise the carrier generated by Bob, which is known as covert communication. Our main contribution is to apply covertness ideas to a quantum-illumination setup to show that it is possible to reach unconditional security. This is achieved by generating a carrier that resembles the thermal noise in the statistical sense so that the signal modulated by Alice cannot be demodulated by a powerful Eve. We discuss that such a system can be implemented when the carrier strength is low and the bandwidth of the signal is wide enough. We provide a step-by-step implementation of these ideas without photodetection, which also solves a known bottleneck in quantum microwave applications.

Alice and Bob can operate on a licensed spectrum since covertness requires them to generate and use low-power carriers. Therefore, the proposed quantum-communication system can accommodate several applications in emerging or well-established areas such as the Internet of Things.

Figure 1

Sketch of the two-way quantum communication protocol. Bob sends a signal to Alice, who embeds the message by phase modulation. The signal is then sent back to Bob, who retrieve the information via a suitable measurement. Bob may use quantum correlations in order to increase the SNR. A passive Eve is able to collect the photons lost in both paths, but she does not have access to Bob and Alice labs.

Figure 2

Setup for the two-way covert quantum communication through a bright bosonic channel. A signal microwave mode ${\hat{a}}_S^{(k)}$, possibly entangled with an idler mode ${\hat{a}}_I^{(k)}$, is generated by Bob. The idler mode is stored in the lab, while the signal is sent through a noisy channel to Alice, who receives the noisy modes ${\hat{a}}_S^{\mathrm{\prime }(k)}$. Alice modulates the phase of the signal by ${\tilde{\phi }}_k=\varphi +{\phi }_k$, where $\varphi$ and ${\phi }_k$ belong to a preagreed discrete alphabet $\mathcal{A}$. Here, $\varphi$ embeds the information to be sent, while $e^{-i{\phi }_k}$ is an encoding operator. The value of ${\phi }_k$ is taken uniformly at random in $\mathcal{A}$, and it is known only to Alice and Bob. The signal is then scattered back to Bob, who decodes it by applying a phase modulation $e^{i{\phi }_k}$. This process is repeated $M$ times ($k=1,\dots ,M$), for each symbol transmission. Bob performs a measurement on the modes $\{e^{i{\phi }_k}{\hat{a}}_R^{(k)},{\hat{a}}_I^{(k)}\}$ in order to discriminate between the possible values $\varphi$. Eve performs a collective measurement on the modes $\{{\hat{w}}_{\leftarrow }^{(k)},{\hat{w}}_{\to }^{(k)}\}$ in order to understand whether Alice and Bob are communicating. If the average power of the signal modes is $O({\eta }^2{N}_B/\sqrt{n})$, then Alice and Bob are able to use covertly $n$ channel modes. This allows transmission of a $O(\sqrt{n})$ number of bits in a secure way. In the $1$–$10$-GHz band, Bob’s signals are generated at 20 mK in order to suppress the thermal contribution and comply with the covertness conditions.

Figure 3

Performance of the quantum-correlated protocol with respect to the idler-free case. (a) Maximal achievable gain of the Chernoff bound of the quantum-correlated case (denoted as ${\beta }_{\max }^{\mathrm{col}}$) with respect to the idler-free case (${\beta }_{\mathrm{cl}}^{\mathrm{col}}$), depending on the average number of thermal photons in the environment $N_B$. For instance, for $N_B=1$, the maximal gain is about $4.6$ dB and it is reached in the $N_S\ll 1$ limit. (b) Comparison of the optimal receiver performance for Gaussian states and Schrödinger’s cat states with respect to the idler-free case, in the $N_B\gg 1$ limit. The performance is quantified in terms of the error probability decaying exponent. Indeed, the quantities $g_{\mathrm{TMSV,\; SC}}^{\mathrm{col}}={\beta }_{\mathrm{TMSV,SC}}^{\mathrm{col}}/{\beta }_{\mathrm{cl}}^{\mathrm{col}}$ and $g_{\mathrm{TMSV,\; SC}}^{\mathrm{loc}}={\beta }_{\mathrm{TMSV,SC}}^{\mathrm{loc}}/{\beta }_{\mathrm{cl}}^{\mathrm{loc}}$ are plotted for $N_B\gg 1$. The graphics show how the advantage in using quantum correlations decays with the transmitting power $N_S$. Gaussian states perform better than Schrödinger’s cat states for finite $N_S$.

Figure 4

Probabilistic version of the two-way covert quantum-communication protocol. Alice and Bob secretly choose to communicate only in the fraction of the temporal and frequency modes at their disposal. In the $on$ setting (green dots), Alice performs a phase modulation $e^{-i(\varphi +\phi )}$, where $\varphi \in \mathcal{A}$ is the symbol that Alice wants to transmit and $\phi$ is taken uniformly at random in $\mathcal{A}$. The phase $\phi$ is secretly known to both Alice and Bob. In the $\mathit{off}$ setting (red dots), Alice performs a phase modulation $e^{-i\phi }$, where $\phi$ is taken uniformly at random in $\mathcal{A}$. Here, $\varphi$ is generated in situ and it is not shared by Alice. This version of the protocol needs $O(\sqrt{n}\log n)$ of shared secret bits to communicate covertly $O(\sqrt{n})$.

Figure 5

Scheme for the preparation of Schrödinger’s cat state. A resonator with central frequency ${\omega }_r$ is driven with a coherent signal, displacing the state of the cavity to $|-i\alpha ⟩$. A transmon qubit with frequency ${\omega }_q$ is initialized in a state $|+⟩=(|e⟩+|g⟩)/\sqrt{2}$. A conditional phase shift is then applied. This is implemented by letting the resonator and qubit interact in the strong dispersive regime for a time $t_{\chi }=\pi /(2\chi )$, where $\chi =g^2/\mathrm{\Delta }$ is the effective coupling. Here, $g$ is the coupling strength of the qubit-resonator system, $\mathrm{\Delta }={\omega }_r-{\omega }_q$ and $g/\mathrm{\Delta }\ll 1$. Finally, a $\pi /2{\hat{\sigma }}_y$ pulse is applied to the qubit. Feasible parameters are ${\omega }_r=5$ GHz, ${\omega }_q={\omega }_r+\mathrm{\Delta }$ with $\mathrm{\Delta }=20$ Mhz, and $g=100$ Khz.

Figure 6

Scheme for the implementation of the observable optimizing the quantum Fisher information. During the signal transmission, which happens at frequency ${\omega }_r$, the qubit state is transferred to a quantum memory. The qubit frequency is then tuned to ${\omega }_q={\omega }_r$. Here the quantum memory is a resonator at frequency ${\omega }_{\mathrm{QM}}$, interacting dispersively with the qubit. This allows the implementation of the cat code. The state is transferred back to the qubit for the measurement stage. The received signal ${\hat{a}}_R$ is attenuated. A squeezing operation is then applied using a Josephson parametric amplifier in the degenerate mode. The output interacts with the qubit in the resonant regime. Finally, the qubit is measured in the $\{|g⟩⟨g|,|e⟩⟨e|\}$ basis.