Quantum Decision Theory for Displacement Detection with Finite-Energy GKP States
We develop a quantum-decision-theoretic framework for detecting phase-space displacements with finite-energy, $d$-level Gottesman-Kitaev-Preskill (GKP) probes. For single-mode and entanglement-assisted architectures, we derive the Bayesian minimum-error probability, the optimal Neyman-Pearson receiver-operating characteristic, and the corresponding minimum detectable displacement. Finite-energy effects are treated through exact theta-series displacement kernels, while pure loss followed by quantum-limited amplification is mapped to an effective Gaussian random-displacement channel. Entanglement removes preparation-dependent blind directions and preserves both logical displacement labels, although it does not surpass the pointwise optimized single-mode strategy in the noiseless pure-state setting. We benchmark the resulting protocols against coherent-state, direction-matched squeezed-vacuum, and twin-beam schemes at equal nominal squeezing. Numerical results identify finite-squeezing and lossy regimes in which GKP probes achieve both a lower Bayesian error and a smaller minimum detectable perturbation than the selected Gaussian receivers.