Performance limits of a quantum receiver for detecting phase-modulated communication signals

TL;DR

This work investigates a quantum receiver for phase-modulated signals, formulating a demodulation protocol based on a generalized cumulant expansion to model noisy quantum dynamics. By deriving the Helstrom-bound-saturating measurements, BEP expressions, and Holevo information, the authors quantify when ensembles of NV-diamond spin sensors can outperform classical Chu-limited electrically-small antennas. The analysis shows that entanglement across qubits yields Heisenberg-like scaling in channel capacity and BEP reduction, while realistic noise (phase and dephasing) can be mitigated via a phase-offset suppression feedback loop. The results indicate practical regimes where quantum receivers can surpass classical limits, especially at lower frequencies and with high defect densities, though engineering challenges such as SPAM time, control fidelity, and decoherence remain significant."

Abstract

Quantum sensors are an ideal candidate for detecting weak electromagnetic signals because of their exceptional sensitivity and compact form factor. In this work, we analyze the performance of a quantum-sensor-based receive chain for demodulating information encoded in phase-modulated electromagnetic waves. We introduce a generalized cumulant expansion to model a noisy quantum receiver and use it to compare the performance of various quantum demodulation protocols. Employing bit error probability (BEP) and channel capacity as quantitative performance metrics, we compare the capabilities of ensembles of quantum sensors - both unentangled and entangled - using Binary Phase-Shift Keying (BPSK) as a representative example of phase modulation. We identify conditions when the channel capacity of an ensemble of quantum sensors may surpass the limits of a classical electrically small antenna. Additionally, we discuss modifications to the quantum protocol that enables high-fidelity data recovery even in the presence of sensor noise and channel distortions. Finally, we explore practical performance limits of such a quantum receive chain, with a focus on NV-diamond as the quantum sensor platform.

Figures (7)

• Figure 1: A quantum-enabled communications scheme. (a) Schematic of a communication system involving a quantum receiver. The transmitter and channel components are classical, while the receiver is quantum. (b) Schematic for a signal demodulation protocol for PSK signals using a quantum receiver. The approach involves synchronizing a quantum sensor to the carrier wave, evolution and measurement of the sensor state; all of these steps must be accomplished within a symbol time. (c) Schematic of a single symbol discrimination: the qubit is prepared in the initial state, the qubit evolves in the presence of the signal, and then it is measured, determining the encoded symbol. Since the protocol utilizes discrimination per symbol, the above must occur within one symbol time, $T_{\textrm{sym}}$.
• Figure 2: Schematic representation of a qubit in the quantum demodulation protocol. A sensor is prepared the ground state $\ket{0}^{\otimes n}$ and measured along the $y$-axis to demodulate the signal. The orange arrow denotes the direction of the signal coupling, the green arrow the control axis, and the purple arrow the initial state. Transverse coupling of the signal requires constant longitudinal control.
• Figure 3: The optimal bit error probability (BEP) occurs for a sensing time inversely proportional to the signal strength, and with a white-noise model, this would be $T_{opt} = \frac{1}{m\Omega_{\rm s}}\tan^{-1}(\frac{\Omega_{\rm s}}{\Gamma})$, where $m$ is the number of entangled qubits, $\Omega_{\rm s}$ is the signal strength, and $\Gamma$ is the dephasing rate. Panel (a) illustrates the BEP (Eq. \ref{['eqn:bep_ensemble']}) as a function of sensing time for three unentangled quantum spins, for multiple levels of $\Gamma$, from $\Gamma=0$ to $2\Omega_{\rm s}$. Panel (b) illustrates the BEP for three quantum spin entangled into a GHZ state. The bottom panel demonstrates that collective sensing reduces the optimal sensing time, as the phase accumulates $m$ times faster, for $m$ spins entangled, but note also, that the GHZ state is more sensitive to the noise environment as well. For slower bit rates, the unentangled ensemble would be superior, but for fast bit rates which limit the sensing time, an ensemble of entangled sensors prove superior.
• Figure 4: Our quantum-enhanced sensing protocol can outperform the limits on classical electromagnetic receivers, because classical receivers become ineffective at small scales. The receiver scale is set to 1 cm and the power flux density to 1.18 $\mathrm{W}/\mathrm{m}^2$, i.e., 100 nT at the quantum receiver. In panel (a), BEP (Eq. \ref{['eqn:bep_ensemble']}) is plotted against the total number of NV centers used in a quantum sensor. Different traces are given for ensembles of quantum sensors initialized into GHZ states of $m$ spins, $\frac{1}{\sqrt{2}}( \ket{+}^{\otimes m}+\ket{-}^{\otimes m})$. We see that entangling the spins reduces the number of spins necessary for a given BEP by an order of magnitude. In panel (b), the channel capacity for classical (Eq. \ref{['eqn:classical_capacity_limit']}) and quantum (Eq. \ref{['eqn:multiqubit_entangled_channel_capacity']}) receivers are plotted against the total number of NV centers used in a quantum sensor. The noiseless quantum channel capacities plotted are the $l>n$ bound (dotted), the QPSK encoding (dashed), and the BPSK encoding (solid). The noiseless quantum channel capacity sets an upper bound on the noisy quantum channel capacity. We find that the crossover between the quantum channel capacity and the Chu-limited channel capacity at 2.4 GHz occurs between one and ten million NV centers at a power flux density of $\sim 1\, \mathrm{W/m^2}$ at the receiver, depending on the amount of entanglement per quantum sensor. The upper bound on the quantum channel capacityoutperforms the Chu limit on classical channel capacity at 10 MHz.
• Figure 5: A phase offset error occurs when the phase of an incoming signal is different from the phase of the 'local oscillator' of the receiver. In a quantum context, this is a phase difference between a signal and the quantum control drive. Panel (a) demonstrates a feedback protocol to correct for a phase offset error in our developed PSK quantum demodulation scheme. The phase error is proportional to $\braket{\sigma^x}$, so by adding fast $Z$ pulses to the dynamics, with a strength proportional to $\braket{\sigma^x}$, one can get the control to be in phase with the incoming signal. Panel (b) demonstrates that this protocol successfully reduces the phase offset, as seen by $\braket{\sigma^x}\to 0$ with the number of feedback cycles. The demodulated signal is encoded in the $\braket{\sigma^y}$ measurement, so the error is reduced as that approaches the true value of $\pm 1$, i.e. encoding a '0' or '1' symbol, respectively. In the plot, the expectation values are calculated with a Monte Carlo simulation of 50 noise trajectories.