AI 中文总结
该研究构建了基于有限能量GKP探针的量子决策理论框架，推导了相关检测性能指标，对比高斯方案后发现其在有限压缩和有损区域检测性能更优。
AI 中文摘要
我们开发了一种量子决策理论框架，用于检测相空间位移，采用有限能量的d能级Gottesman-Kitaev-Preskill（GKP）探针。针对单模和纠缠辅助架构，我们推导了贝叶斯最小错误概率、最优奈曼-皮尔逊接收机工作特性以及对应的最小可检测位移。通过精确的θ级数位移核处理有限能量效应，将纯损耗后接量子受限放大映射为有效高斯随机位移信道。纠缠消除了依赖制备的盲方向并保留了两个逻辑位移标签，不过在无噪声纯态场景下，它并未超越逐点优化的单模策略。我们在名义压缩度相等的情况下，将所得协议与相干态、方向匹配压缩真空及孪生光束方案进行基准测试。数值结果表明，在有限压缩和有损区域，GKP探针相较于所选高斯接收机，兼具更低的贝叶斯误差和更小的最小可检测扰动。
英文摘要
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.
