The transfer operator KO maps alterations in historical measurement records to modifications in prospective conditional observables. For polynomial bosonic systems subject to continuous quadrature monitoring, this analysis reveals a distinct boundary delineating finite and infinite realizability. Gaussian dynamics and the Conditional Momentum Moment class enable precise finite-dimensional filters.

Conversely, nonlinear dynamics falling outside these classifications typically generate an indefinite number of independent response directions and preclude key finite-dimensional C1 observable or state-level filtering. The singular values of KO subsequently broaden the realization framework to encompass approximation, yielding the optimal rank-d local-response error exceeding that achievable within those finite classes.

Quantifying limitations in reconstructing quantum states from continuous weak measurements

Numerical experiments with 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. These results identify the limit of exact quantum filtering and quantify finite-dimensional approximation beyond it. Continuous quantum measurement describes systems whose evolution is conditioned on a continuously acquired measurement record; each record produces a different stochastic trajectory through combined open-system dynamics and measurement backaction. Such trajectories are central to quantum state estimation, feedback, sensing, and measurement-based quantum information processing.

In several important settings, notably linear, Gaussian systems, conditional dynamics are described exactly by finitely many evolving variables. Exact descriptions also exist when Heisenberg dynamics are linear or conditional states remain within record-independent finite-dimensional families. Nonlinear monitored systems usually generate infinite moment hierarchies, but the failure of one hierarchy does not guarantee overall complexity as alternative coordinates might still close.

Therefore, whether any finite-dimensional causal realization exists was addressed by viewing filtering as a past-to-future problem. For an observable O differentiating between the past record and future trajectory of ⟨O⟩c generates K_O, measuring how changes to the past affect the future observable. Its rank determines exact realizability while singular values measure accuracy at chosen dimensions; this follows classical input, output theory where inputs are distinguished through effects on outputs characterised via Hankel maps.

Related approaches to quantum filter reduction include estimation-algebra methods, exact reductions preserving selected observables, and explicit low-dimensional descriptions for systems with suitable invariant structure. These techniques identify or construct reduced filters when specific structures exist. A test of the record-to-observable response is performed directly without first choosing coordinates or form of a reduced filter; this contribution has three parts.

First, an observable-level realization theory for quantum filtering is developed. Any strong d-dimensional C1 filter forces K_O to have rank at most d. Infinite rank rules out every such finite realization of the chosen observable regardless of its coordinates, also preventing an exact finite realization of the full conditional state. Second, applying this framework to polynomial bosonic dynamics yields an exact classification for one continuously monitored mode.

The Gaussian and Conditional Momentum Moments (CMM) families are precisely those polynomial Hamiltonians admitting finite exact filters; all other Hamiltonians exhibit infinite observable response rank for almost every Gaussian preparation. This boundary remains unchanged by inefficient detection or thermal noise and persists when the monitored mode resides within a finite multimode network. For polynomial bosonic systems under continuous quadrature monitoring, exact finite-dimensional descriptions exist in some cases but aren’t generally possible; there isn’t a universal criterion determining which systems allow such filters.

A linearized observable Hankel operator K_O maps changes in past measurement records to future conditional observables. The results show that exact finite-dimensional descriptions exist for some continuously monitored quantum systems, yet no general criterion separates those admitting such filters from others.

This question is formulated as a realization problem introducing the linearized observable Hankel operator K_O, mapping changes in past records to future observables. Dynamics belonging to either the Gaussian or Conditional Momentum Moment class allow exact finite-dimensional filters; however, nonlinear dynamics beyond these classes generally produce an infinity of independent response directions preventing robust filtering.

The approach begins with classical input, output realization theory then introduces the phase-space model considered here. Next, closures for both Gaussian and CMM are described before formulating realization theory at the level of individual observables. Causality dictates that zt, and therefore bOt, depends only on records up to time t; future increments aren’t available to the filter.

Finite-dimensional filters represent all needed record information for those outputs using finite internal coordinates. The C1 assumption ensures differentiable dependence on record perturbations within an open neighborhood and ‘strong’ means this holds across such neighborhoods rather than at isolated points. This viewpoint applies to any input, output stochastic system asking whether dependence on the input record can be represented by finitely many evolving variables without altering desired outputs.

In quantum filtering, the record is a continuously acquired measurement signal from one experimental run while observables represent conditional expectations like ⟨O⟩t. Once constructed, filters become coupled equations driven by observed trajectories Yt; each experiment realization sees its record feed into the filter propagating coordinates zt causally before reading out observable trajectory through q(zt, t). These coordinates need not have prescribed physical meaning, they may be moments, Fock coefficients or convenient nonlinear variables unrelated to standard state representations.

Their role is exactly propagating information needed to produce desired output. Realization theory studies when such input, output maps admit finite internal states providing coordinate-independent language used below to prove their main result.

The approach next specifies quantum input and output model applying this realization framework. Input record represents homodyne or heterodyne measurement current while outputs are conditional expectations of system observables; an additional subscript has been suppressed throughout classification as all expectations and states are conditioned on the observed record.

Exact finite-dimensional descriptions exist for some continuously monitored quantum systems but no general criterion separates those that admit such filters from others. For polynomial bosonic systems under continuous quadrature monitoring, this provides a boundary between finite and infinite realizability.

Gaussian dynamics and the Conditional Momentum Moment class allow exact finite-dimensional filters; nonlinear dynamics outside these classes generally produce infinitely many independent response directions preventing any robust finite-dimensional filter. Continuous quantum measurement describes systems evolving based on continuously acquired records creating stochastic trajectories from open-system dynamics and measurement backaction.

Such trajectories are central to quantum state estimation, feedback, sensing, and information processing. Nonlinear monitored systems often generate infinite moment hierarchies; however failure of one hierarchy does not guarantee overall complexity as alternative coordinates might still close.

The basic question is whether any finite-dimensional causal realization exists which can be addressed by viewing filtering as a past-to-future problem. Its rank determines exact realizability while singular values measure accuracy at chosen dimensions; this follows classical input, output theory where inputs are distinguished through effects on outputs characterised via Hankel maps.

Exact finite-dimensional descriptions exist for some continuously monitored quantum systems, yet no general criterion separates those that admit such filters from others. Gaussian dynamics and the Conditional Momentum Moment class allow exact finite-dimensional filters; nonlinear dynamics outside these classes generally produce infinitely many independent response directions preventing any robust finite-dimensional filter.

Quantifying simplifiability unlocks advances in scalable quantum technologies

Researchers at Princeton University have devised a way of classifying quantum systems based on whether they can be described using a limited number of variables, an important step towards streamlining calculations in fields like sensing and cryptography. However their approach reveals fundamental tension: identifying when simplified descriptions are possible is now within reach while determining the degree of simplification remains challenging.

The research identified that polynomial bosonic systems undergoing continuous quadrature monitoring possess a clear boundary between those allowing exact finite-dimensional filters and those which do not. This matters because it establishes criteria for simplifying complex quantum system descriptions, potentially easing computational demands. Specifically, Gaussian dynamics and the Conditional Momentum Moment class admit these simpler representations, whereas more general nonlinear dynamics typically require infinitely many variables to describe accurately. Researchers used the linearized observable Hankel operator K_O to quantify this simplifiability and assess accuracy in reduced models.

👉 More information
🗞 Realization Theory for Quantum Filtering: Classifying Continuously Monitored Bosonic Systems
✍️ Jacob Emerson
🧠 ArXiv: https://arxiv.org/abs/2609.09663

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