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

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 $W₂$ as a robust geometric metric for lossy quantum sensing. Unlike overlap-based measures, W₂ 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 W₂ 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.