Realization Theory for Quantum Filtering: Classifying Continuously Monitored Bosonic Systems
Exact finite-dimensional descriptions exist for a few important classes of continuously monitored quantum systems, but there is no general criterion separating systems that admit such filters from those that do not. We formulate this question as a realization problem and introduce the linearized observable Hankel operator $K_O$, which maps changes in the past measurement record to changes in a future conditional observable. For polynomial bosonic systems under continuous quadrature monitoring, this gives a sharp boundary between finite and infinite realizability. Gaussian dynamics and the Conditional Momentum Moment class admit exact finite-dimensional filters, while nonlinear dynamics outside these classes generically produce infinitely many independent response directions and admit no robust finite-dimensional $C^1$ observable- or state-level filter. The singular values of $K_O$ then extend the same realization framework to approximation, giving the optimal rank-$d$ local-response error beyond the finite classes. Numerical experiments with the Kerr and Duffing oscillators, together with a dynamics-adapted Poisson--Charlier construction for Kerr dynamics, show how this observable-level benchmark interacts with concrete reduced representations. Together, these results identify the boundary of exact quantum filtering and quantify finite-dimensional approximation beyond it.