发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出线性化可观测量汉克尔算子，为连续监测玻色子系统划分精确量子滤波的边界，并量化有限维近似的误差。
AI 中文摘要
对于几类重要的连续监测量子系统，存在精确的有限维描述，但尚无一般性判据来区分哪些系统允许此类滤波器，哪些不允许。我们将此问题表述为实现问题，并引入线性化可观测量汉克尔算子$K_O$，该算子将过去测量记录的变化映射到未来条件观测量的变化。对于连续正交监测下的多项式玻色子系统，这给出了有限可实现性与无限可实现性之间的清晰边界。高斯动力学和条件动量矩类允许精确的有限维滤波器，而属于这些类之外的非线性动力学通常会产生无限多个独立响应方向，且不存在鲁棒的有限维$C^1$可观测量级或状态级滤波器。$K_O$的奇异值随后将同一实现框架扩展到近似领域，给出了有限类之外的最优秩-$d$局部响应误差。利用Kerr和Duffing振荡器进行的数值实验，以及针对Kerr动力学构建的动力学自适应Poisson-Charlier方法，展示了这一可观测量级基准如何与具体降维表示相互作用。综合来看，这些结果确定了精确量子滤波的边界，并量化了超越该边界的有限维近似。
英文摘要
Nonlinear dynamics generate non-Gaussian states and operations, but they also make continuously monitored quantum systems difficult to track. A quantum filter performs this task by compressing a noisy measurement record into a set of evolving variables that predict future observables. For Gaussian dynamics, this compression is exact and finite. In nonlinear systems, conventional moment equations generally expand without bound, but the failure of one representation does not prove that every finite description is impossible. Here we formulate quantum filtering as a realization problem for general quantum input--output maps. We introduce the linearized observable Hankel operator $K_O$, which maps changes in the past measurement record to changes in a future conditional observable. Its rank provides a coordinate-independent test of whether finite compression is possible. Applied to single-mode polynomial bosonic systems under continuous quadrature monitoring, this test yields a sharp boundary: Gaussian dynamics and the measurement-aligned Conditional Momentum Moment class admit finite realizations, whereas dynamics outside these classes produce infinitely many response directions and admit no finite-dimensional smooth exact filter. Exact impossibility, however, need not imply large practical complexity. The singular values of $K_O$ quantify how many response directions matter at a chosen local accuracy, defining the effective memory of an observable. For the Kerr and Duffing oscillators, these singular values decay rapidly despite infinite exact rank, revealing strong local compressibility while showing substantial room to improve existing global filters. The framework therefore separates exact realizability from effective complexity in monitored nonlinear quantum dynamics.
Comments25 pages, 5 figures
