Quantum illumination for enhanced detection of Rayleigh-fading targets

Quantum illumination (QI) is an entanglement-enhanced sensing system whose performance advantage over a comparable classical system survives its usage in an entanglement-breaking scenario plagued by loss and noise. In particular, QI's error-probability exponent for discriminating between equally likely hypotheses of target absence or presence is 6 dB higher than that of the optimum classical system using the same transmitted power. This performance advantage, however, presumes that the target return, when present, has known amplitude and phase, a situation that seldom occurs in light detection and ranging (lidar) applications. At lidar wavelengths, most target surfaces are sufficiently rough that their returns are speckled, i.e., they have Rayleigh-distributed amplitudes and uniformly distributed phases. QI's optical parametric amplifier receiver—which affords a 3 dB better-than-classical error-probability exponent for a return with known amplitude and phase—fails to offer any performance gain for Rayleigh-fading targets. We show that the sum-frequency generation receiver [Zhuang et al., Phys. Rev. Lett. 118, 040801 (2017)]—whose error-probability exponent for a nonfading target achieves QI's full 6 dB advantage over optimum classical operation—outperforms the classical system for Rayleigh-fading targets. In this case, QI's advantage is subexponential: its error probability is lower than the classical system's by a factor of $1/ln(M\overline{\kappa }{N}_S/N_B)$, when $M\overline{\kappa }{N}_S/N_B\gg 1$, with $M\gg 1$ being the QI transmitter's time-bandwidth product, $N_S\ll 1$ its brightness, $\overline{\kappa }$ the target return's average intensity, and $N_B$ the background light's brightness.

Figure 1

Schematic representation of the SFG receiver's $k\mathrm{th}$ stage, showing only the $m\mathrm{th}$ mode pair, although all $M$ mode pairs are processed simultaneously. The $m\mathrm{th}$ mode pair of the returned light (${\widehat{a}}_{R_m}^{\left(k\right)}$) and the retained idler (${\widehat{a}}_{I_m}^{\left(k\right)}$) at the input to the $k\mathrm{th}$ stage is transformed into the corresponding mode pair at that stage's output by means of SFG. Photon-counting measurements are made on the single-mode sum-frequency output (${\widehat{b}}^{\left(k\right)}$) and the auxiliary output modes ($\{{\widehat{a}}_{E_m}^{\left(k\right)}:1\le m\le M\}$). The SFG receiver's decision as to target absence or presence is based on the total of all the photon-counting measurements, i.e., $N_T\equiv {\sum }_{k=1}^K(N_b^{\left(k\right)}+N_E^{\left(k\right)}),$ where $N_b^{\left(k\right)}$ is the outcome of the ${\widehat{b}}^{\left(k\right)†}{\widehat{b}}^{\left(k\right)}$ measurement, and $N_E^{\left(k\right)}$ is the outcome of the ${\sum }_{m=1}^M{\widehat{a}}_{E_m}^{\left(k\right)†}{\widehat{a}}_{E_m}^{\left(k\right)}$ measurement.

Figure 2

QI and CI ROCs for Rayleigh-fading target detection with $\overline{\kappa }=0.01$, $N_B=20$, and $\epsilon =0.01$. (a) $N_S={10}^{-4}$ and $M={10}^{8.5}$. (b) $N_S={10}^{-2}$ and $M={10}^{6.5}$.

Figure 3

QI and CI error probabilities for Rayleigh-fading target detection with ${\pi }_0={\pi }_1=1/2$, $\overline{\kappa }=0.01$, $N_B=20$, and $\epsilon =0.01$. (a) $N_S={10}^{-4}$. (b) $N_S={10}^{-2}$. The slope discontinuity in $Pr{\left(e\right)}_{\mathrm{SFG}}$ for $N_S={10}^{-2}$ is due to its receiver's photon-number threshold increasing from $n_t=0$ to $n_t=1$ at that point.