Finite Realizations and Effective Memory in Nonlinear Quantum Filtering

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. Any finite-dimensional smooth filter forces K O to have finite rank, so infinite rank provides a coordinate-independent proof that no exact finite-dimensional filter exists. Applied to single-mode polynomial bosonic systems under continuous quadrature monitoring, this 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, showing that infinite exact memory can coexist with low finite-accuracy complexity. The framework therefore separates exact realizability from effective complexity in nonlinear quantum filtering.