From State-Space Transport to Measurement-Aware Distinguishability in Quantum Sensing

Abstract

Overlap-based distinguishability measures, such as fidelity- or Chernoff-type quantities, play a central role in quantum sensing and quantum illumination. In strongly lossy and fluctuating environments, however, these quantities may become numerically compressed and therefore less informative for optimization, monitoring, or adaptive control. In this work, we investigate transport-based distinguishability criteria for lossy quantum sensing. We first introduce an isotropic Gaussian transport metric defined on first and second moments and compare it with a fidelity-based benchmark in a thermal-loss model. We then show analytically that, within an isotropic thermal-reference geometry, this metric locally disfavors squeezing relative to coherent displacement, thereby distinguishing global phase-space robustness from directional metrological advantage. We next introduce a projected transport metric adapted to quadrature-resolved measurements and show that its optimization over the measurement quadrature is analytically tractable, reducing to a boundary choice between the principal axes of the output noise ellipse. We further extend the framework to a measurement-aware metric defined on detector output statistics, and derive an explicit Gaussian formula for a noisy quadrature measurement chain. Finally, in a fading setting, we show that the isotropic metric and the projected metric aligned with the coherent displacement retain first-order sensitivity to the transmissivity in the strong-loss regime, whereas the orthogonal projected metric is compressed to second order. These results support a hierarchical view of transport-based distinguishability in quantum sensing, ranging from global robustness indicators to measurement-adapted operational metrics.