Channel Superposition Mitigates Photon Loss Errors in Quantum Illumination

TL;DR

The paper tackles the challenge of photon loss eroding the quantum advantage in quantum illumination by introducing a channel-superposition framework that coherently combines two noisy-channel realizations. It develops two concrete protocols, indefinite causal order (ICO) and path superposition with disjoint environments (PS-DE), and analyzes their performance using the quantum Chernoff bound, showing that any nonzero interference yields an advantage over standard QI, with ICO offering greater robustness due to shared environmental coupling. The authors provide analytical bounds tying the potential gain to interference terms and spectral norms, and corroborate these findings with numerical simulations demonstrating ICO’s superior Chernoff exponents, especially in high-loss, low-reflectivity regimes. They also formulate a resource-theoretic view of the coherence required for channel superposition, show robustness to control-qubit noise (bit and phase flips), and compare ICO to a coherent-environment control scheme (DCO), concluding that ICO achieves similar benefits with reduced experimental overhead. The work suggests practical routes to implement ICO-based quantum illumination and outlines future directions, including scaling to more channels and applying the approach to other quantum sensing tasks.

Abstract

In quantum illumination, the probe photon is entangled with an ancilla photon, and both are jointly measured at the end. The entanglement between the probe and ancilla photons enhances the detection performance per unit average photon number in the probe mode, particularly in low-reflectivity and high-noise scenarios. However, photon loss severely limits the practical advantage of such protocols. To address this, we employ a channel superposition framework, which encompasses two kinds of channel superposition protocols: indefinite causal order (ICO) and path superposition with disjoint environment (PS-DE). Our analytical and numerical analysis based on the quantum Chernoff bound shows that both ICO and PS-DE can, in principle, achieve an advantage. The advantage persists as long as non-zero interference remains, reverting to the performance of standard quantum illumination once the interference is completely suppressed. Crucially, the ICO protocol is significantly more robust, maintaining a tighter upper bound on the error probability than standard quantum illumination and the PS-DE approach. This performance hierarchy is rooted in their fundamental structures: ICO exploits a shared environment to generate stronger quantum interference, while PS-DE, relying on disjoint environments, offers a more experimentally tractable albeit less potent alternative.

Figures (10)

• Figure 1: Quantum illumination protocols under photon loss: QI, ICO, and PS-DE.Key distinction: Both ICO and PS-DE introduce quantum interference in the output state to enhance noise resilience relative to standard QI. ICO superposes causal orders through a shared environment, while PS-DE superposes spatial paths via disjoint, independent environments. (a–b) Standard QI protocol. (a): mode $A$ interacts with the target $\mathcal{E}^{(E)}_\eta$, where it undergoes probabilistic photon loss, represented by $\mathcal{E}^{(D)}$ before and $\mathcal{E}^{(F)}$ after the detection event. Mode $B$ is retained as an ancilla for subsequent joint measurements. (b): A schematic of the sequential process. Mode $A$ passes through $\mathcal{E}^{(D)} \to\mathcal{E}^{(E)}_{\eta}\to\mathcal{E}^{(F)}$, followed by a joint measurement (POVM) together with mode $B$. (c–d) ICO protocol. (c): Mode $A$ propagates through one of two paths selected by a Mach-Zehnder interferometer (MZI), with the control system modulating the phase shift by the state of the control qubit $\ket{\psi_c}$. If the state of the control qubit is $\ket{\psi_c}=\ket{0}$ (corresponding to path $A^{(0)}$), or $\ket{\psi_c}=\ket{1}$ (corresponding to path $A^{(1)}$), mode $A$ experiences photon losses in reverse orders, thereby enabling exploration of causal indefiniteness. Crucially, the two paths undergo coherent recombination at the end of the MZI. (d) Schematic of the ICO protocol: The sequence of channel interactions for mode $A$ ($\mathcal{E}^{(D)}$, $\mathcal{E}^{(E)}_\eta$ and $\mathcal{E}^{(F)}$) is dictated by the state of the control qubit $\ket{\psi_c}$. (e–f) PS-DE protocol. (e): Mode $A$ undergoes path superposition via a controlled MZI, where each path experiences disjoint channel combinations (e.g., $A^{(0)}$ encounters $\mathcal{E}^{(D)}$ and $\mathcal{E}^{(Y)}$; $A^{(1)}$ encounters $\mathcal{E}^{(F)}$ and $\mathcal{E}^{(X)}$) before coherent recombination. (f) Schematic of the PS-DE protocol: the channel processing for mode $A$ leverages quantum interference from superposing distinct paths/channels.
• Figure 2: Schematic diagram of a unitary circuit. (a)ICO protocol. Photon A interacts with three auxiliary modes (D, E, F) via unitary operators $V_{AD}$, $V_{AE}$, $V_{AF}$, with causal order controlled by qubit $c$. When the control qubit $c=\ket{0}$, the photon travels through paths D, E, and F in sequence (i.e., following the order D $\to$ E $\to$ F); when $c=\ket{1}$, the photon travels through paths F, E, and D in reverse order (i.e., following the order F $\to$ E $\to$ D). (b)PS-DE protocol. Photon A interacts with five ancilla modes (D, F, E, X, Y) through unitary operators $V_{AD}$, $V_{AF}$, $V_{AE}$, $V_{AX}$, and $V_{AY}$. When the control qubit $c=\ket{0}$, the photon traverses paths D, E, and Y; when $c=\ket{1}$, the photon traverses paths F, E, and X.
• Figure 3: Impact of Truncated Dimension on $\epsilon$ for QI, PS-DE, and ICO Protocols. This figure demonstrates the relationship between the truncated dimension $\mathcal{D}$ and the performance metrics $\epsilon^{\text{QI}}$, $\epsilon^{\text{PS-DE}}$, and $\epsilon^{\text{ICO}}$. The truncated dimension represents the maximum number of states considered in a quantum system, serving as an approximation technique to manage computational complexity in quantum simulations. As $\mathcal{D}$ increases, $\epsilon^{\text{QI}}$, $\epsilon^{\text{PS-DE}}$, and $\epsilon^{\text{ICO}}$ converge, highlighting the diminishing impact of truncation on simulation accuracy at higher dimensions. The simulation parameters include a target reflectivity $\eta = 0.1$, an average photon number $N=0.5$ for mode $E$, an invariant loss channel probability $p = 0.8$, and an average photon number $N_t=0.01$ for the transmitted mode $A$.
• Figure 4: ICO Protocol Generates Stronger Quantum Interference Than PS-DE Protocol. The ratio $||\sigma^{\text{PS-DE}}||{\text{spec}} / ||\sigma^{\text{ICO}}||{\text{spec}}$ is plotted against the survival probability $p$. The ratio increases from 0 to approximately 0.8, with its slope growing with $p$. The curves for different target reflectivities ($\eta = 0.01, 0.05, 0.1$) are nearly indistinguishable, indicating the ratio is robust against changes in $\eta$. These results demonstrate that the ICO protocol generates a consistently stronger quantum interference effect than the PS-DE protocol across a wide range of parameters.
• Figure 5: Numerical simulation results showing the Chernoff exponent $\epsilon$ for QI, PS-DE, and ICO protocols as a function of survival probability $p$. Results are obtained at fixed truncation dimension $\mathcal{D}=10$ and for reflectivities $\eta=0.01, \eta=0.05$, and $\eta=0.1$. The ICO protocol exhibits superior performance for $p \lesssim 0.6$--$0.7$, beyond which the PS-DE protocol performs comparably.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Proof 1
• Proof 2
• Proof 3
• Proof 4