Quantum Ranging with Gaussian Entanglement

It is well known that entanglement can benefit quantum information processing tasks. Quantum illumination, when first proposed, was surprising as the entanglement’s benefit survived entanglement-breaking noise. Since then, many efforts have been devoted to study quantum sensing in noisy scenarios. The applicability of such schemes, however, is limited to a binary quantum hypothesis testing scenario. In terms of target detection, such schemes interrogate a single spatiotemporal resolution bin at a time, limiting the impact to radar detection. We resolve this binary-hypothesis limitation by proposing an entanglement-assisted quantum ranging protocol. By formulating a ranging task as a multiary hypothesis testing problem, we show that entanglement enables a 6-dB advantage in the error exponent against the optimal classical scheme. Moreover, the proposed ranging protocol can also be used to implement a pulse-position modulated entanglement-assisted communication protocol. Our ranging protocol reveals entanglement’s potential in general quantum hypothesis testing tasks and paves the way toward a quantum-ranging radar with a provable quantum advantage.

Figure 1

Schematic of the entanglement-assisted ranging protocol. In (a), the signal mode ${\widehat{a}}_S$ (blue) and the idler mode ${\widehat{a}}_I$ (red) are initially entangled in a TMSV state. The signal is sent out to probe the range of a target with reflectivity $\kappa$. When the target is at distance $h\mathrm{\Delta }$, the mode ${\widehat{a}}_h$, highlighted in red, collected at time $t_h=2h\mathrm{\Delta }/c$, contains the reflection from the target embedded in noise, while the rest of the collected modes (orange) contain entirely noise. Subplot (b) shows the $m$ possible states in the hypothesis testing problem at the receiver side. In each case, the idler (blue) is correlated with the reflected mode (red).

Figure 2

Error probability performance versus the number of modes $M$ of the quantum ranging protocol compared to the classical schemes. Signal brightness $N_S=0.001$ and target reflectivity $\kappa =0.01$. The number of range slices $m$ and the environmental noise $N_B$ are chosen as (a) $m=2,N_B=3$, (b) $m=3,N_B=1$, and (c) $m=50,N_B=20$. For the entangled strategy, we evaluate the asymptotically tight quantum Chernoff bound (QCB) $P_{E,H}$ (orange dashed), and an exact upper bound $P_{E,\mathrm{UB}}$ (red dashed). As an upper bound, $P_{E,\mathrm{UB}}$ has values above one in certain region of $M$, where the unity bound of probability dominates. For the classical strategy, we evaluate the QCB $P_{C,H}$ (green dashed), an exact lower bound $P_{C,\mathrm{LB}}$ (black dashed), and the coherent-state direction detection performance $P_{C,\mathrm{DD}}$ (black solid). In (a), we also present the optical-parametric-amplifier-based receiver performance (red solid) in an entangled strategy and the numerical results of the classical Helstrom limit (purple star). In (a),(b), the performance of the pretty good measurement (PGM) for coherent-state inputs is also evaluated numerically compared to the classical QCB. The error bars are from estimation of errors due to the finite photon number cutoff in numerical calculations.

Figure 3

(a) Schematic of the entanglement-assisted communication protocol. (b) Information rate of the entanglement-assisted communication protocol with $\kappa =0.1$ and $N_B=20$. We compare the optimized rates $R_M^{\star }$ for the fixed number of repetition modes $M={10}^2,{10}^3,{10}^4$ (red, purple, blue) and the entanglement-assisted classical capacity (black) versus the signal average brightness $n_S$.