Geometric Resilience of Quantum LiDAR in Turbulent Media: A Wasserstein Distance Approach

Abstract

Quantum-enhanced LiDAR, exploiting squeezed states of light, promises significant sensitivity gains over classical protocols. However, in realistic scenarios characterized by high optical losses and atmospheric turbulence, standard figures of merit, such as quantum fidelity or the quantum Chernoff bound, saturate rapidly, failing to provide a usable gradient for system optimization. In this work, we propose the Quantum Wasserstein Distance of order 2 ($W_2$) as a robust geometric metric for lossy quantum sensing. Unlike overlap-based measures, $W_2$ quantifies the transport cost in phase space and maintains a linear response to channel transmissivity, even in regimes where the quantum state is virtually indistinguishable from thermal noise. We derive an analytical threshold for the quantum advantage, demonstrating that squeezing is only beneficial when the transmissivity exceeds a critical value determined by the environmental noise-to-signal ratio. Furthermore, using Monte-Carlo simulations of a fading channel, we show that $W_2$ acts as a high-fidelity estimator of instantaneous link quality, exhibiting a wide dynamic range immune to the numerical instabilities of fidelity-based metrics. This geometric framework bridges the gap between quantum optimal transport and practical receiver design, paving the way for adaptive sensing in scattering media.

Figures (4)

• Figure 1: Benchmarking distinguishability metrics in a high-loss environment ($N_{\text{tot}}=5, n_{\text{th}}=2$). The Quantum Bhattacharyya Bound ($\xi_{\text{QBB}}$, dashed red) saturates rapidly, indicating a vanishing state overlap that hinders gradient-based optimization. In contrast, the Wasserstein distance ($W_2^2$, solid blue) maintains a robust linear behavior, providing a continuous "transport cost" gradient even in regimes where the quantum states are nearly indistinguishable by overlap.
• Figure 2: Map of the optimal resource allocation strategy for a fixed energy budget ($N_{\text{tot}}=10$) in a low-noise environment ($n_{\text{th}}=0.1$). The color gradient represents the detection score ($W_2^2$). The white dashed line tracks the optimal squeezing fraction $\lambda_{\text{opt}}$. We observe a phase transition: for $\eta > 0.25$, the system enters a "Quantum Regime" where maximizing squeezing ($\lambda \to 0.95$) is beneficial. Below this threshold, the optimal strategy collapses to the "Classical Regime" ($\lambda \to 0$), as the fragility of squeezed states to loss outweighs their geometric advantage.
• Figure 3: Parametric analysis of the optimal squeezing fraction versus channel transmissivity. The transition from classical ($\lambda=0$) to quantum ($\lambda > 0$) strategies is dictated by the thermal noise floor. In high-noise environments ($n_{\text{th}}=2.0$, black dashed line), the quantum advantage is suppressed until $\eta > 0.5$. Reducing the noise ($n_{\text{th}}=0.1$, blue solid line) or increasing the signal power ($N_{\text{tot}}=20$, red line) significantly expands the operational range of the quantum regime.
• Figure 4: Statistical distribution of detection metrics in a turbulent fading channel (Monte Carlo simulation, $10^4$ realizations). The inset shows the probability distribution of the channel transmissivity $\eta$. The Wasserstein metric (blue) exhibits a wide dynamic range, mirroring the channel fluctuations. In contrast, the fidelity-based metric (red) is compressed into a narrow spike due to logarithmic saturation. This contrast highlights that $W_2$ acts as a sensitive estimator of instantaneous link quality, whereas $\xi_{\text{QBB}}$ flattens the information.