Single-shot detection limits of quantum illumination with multi-qudit states

Abstract

Quantum illumination is a protocol for detecting a low-reflectivity target by using two-mode entangled states composed of signal and idler modes, which can outperform unentangled states. We study multi-qudit states for single-shot detection limits of quantum illumination under white noise environment. Using three-qubit states, we obtain that the performance is enhanced by the entanglement between signal and idler qubits, whereas it is degraded by the entanglement between signal qubits. The similar behaviors are also observed for three-qutrit, four-qubit, and four-ququart states. In particular, the optimal state is not a maximally entangled multipartite state but a combination of a maximally entangled bipartite state. Moreover, we show that quantum correlation can explain the quantum advantage of three-qubit, three-qutrit, and four-qubit states, with exception of a four-ququart state.

Figures (11)

• Figure 1: A schematic of QI with three modes in order to determine whether there is a target $(\eta\neq0)$ or not $(\eta=0)$, where $\eta$ is a target reflectivity. There are three configurations: (a) two-signal and one-idler ($2S1I$) modes , (b) one-signal and two-idler ($1S2I$) modes, and (c) three-signal ($3S$) modes. The modes in the dashed circles can be entangled or not. A separable state is the classical benchmark in the scenario.
• Figure 2: Three configurations with three-qubit states. Red (blue) circle represents signal (idler) qubit. For each configuration, entangled qubits are demonstrated with gray color.
• Figure 3: Hypothesis testing involving null hypothesis $(H_0)$ and alternative hypothesis $(H_1)$. The probability of detecting $H_1$ when $H_0$ is actually true is called false-alarm probability (Type I error), and the probability of detecting $H_0$ when $H_1$ is actually true is called miss-detection probability (Type II error). The horizontal axis represents a measurable parameter, and the vertical axis represents a probability.
• Figure 4: HB of S-SI state in $2S1I$ configuration as a function of $\eta$ and $p_0$. The dashed lines represent the boundaries of the HB. R1 and R2 represent non-illuminable regions, whereas R3, R4, and R5 represent illuminable regions.
• Figure 5: (a) Mean HB for $2S1I$ configuration, which is summarized in a table. SS-I, GHZ, and S-SI states can be converted to S-S-I state by a single parameter (dashed line). W-state has detection error probability between the S-SI state and the SS-I state with two parameters (blue zone). (b) HB as a function of $\eta \in [0,0.01]$ at $p_0=0.5$, where the order of the HB is S-SI<W<GHZ=S-S-I<SS-I.