Rigorous convergence theory for controlled discrete quantum filtering and feedback models as the time step vanishes

The discrete quantum-binomial framework of Bouten, van Handel and James (SIAM Review, 2009) develops quantum filtering and feedback control on a finite-dimensional model of repeated interactions between a qubit and a chain of field time slices, and asserts — without proof, citing literature that covers only uncontrolled models — that as the interaction time $\lambda^2$ per slice tends to zero (with the number of slices $k$ satisfying $T = k\lambda^2$ for fixed horizon $T$) the controlled models and quantum filters converge to the continuous-time theory of Hudson–Parthasarathy quantum stochastic differential equations and the controlled Belavkin filter; the paper derives the discrete dynamic-programming (Bellman) equation in analogy with continuous-time optimal control but does not assert convergence of value functions or optimal policies. Later work closed the uncontrolled case in full generality, including strong convergence (Bouten–van Handel 2008; Belton–Gnacik–Lindsay 2018), and the controlled case only in restricted forms: convergence in law for a two-level atom with open-loop and Markovian state feedback (Pellegrini 2009), repeated quantum non-demolition measurements without control in the loop (Bauer–Benoist–Bernard 2013), and strong convergence for coherent-feedback networks whose loop carries no measurement record (Bouten–Gough 2026). Still open is the general controlled case: convergence, as $\lambda \to 0$, of the closed-loop filter for repeated-interaction models in the full Hudson–Parthasarathy class under measurement-based (separated) feedback, and convergence of the associated Bellman value functions and optimal separated policies to the continuous-time partially observed control problem. The problem asks for this theory, for a counterexample within the class, or for a rigorous theory on a precisely delimited subclass.

Background

Quantum filtering is the quantum analogue of nonlinear filtering: an open quantum system (the 'plant') is coupled to its environment, the environment is monitored continuously, and one recursively estimates the system's conditional state — a density matrix on a Hilbert space — from the classical stream of measurement outcomes. The estimating recursion is the quantum filter (Belavkin's filter, the ancestor of the stochastic master equation used throughout quantum physics). When the estimate is used to steer the system, one obtains quantum feedback control; the physically complete Markovian architecture for measurement-based control is the separated controller: the control applied at a given step is a function of the filter state, which itself depends on the whole realized measurement record. The continuous-time theory lives on Bosonic Fock space in the Hudson–Parthasarathy (HP) quantum stochastic calculus, where the system evolves under a quantum stochastic differential equation (QSDE) driven by creation, annihilation, and gauge processes, and the controlled filter is a stochastic differential equation driven by the measurement output.

The source paper (Bouten–van Handel–James 2009) constructs the same theory using only finite-dimensional linear algebra. The field is a chain of k independent two-level time slices; each slice interacts with the system qubit for a time $\lambda^2$, with $T = k\lambda^2$ fixed. This is the repeated-interaction (collision) model of quantum optics. Three discrete noise processes — quantum binomial counting processes, the natural two-state analogues of the Poisson and Gaussian noises — play the roles of the three fundamental HP noises. Every continuous-time object — the unitary dynamics, the nonlinear filter, the Zakai-type linear filter obtained through a discrete change of state (discrete Girsanov), the Bellman equation of optimal control, and Lyapunov (supermartingale) design — appears first as an exact finite-dimensional discrete object; for the dynamical and filtering objects the paper claims convergence to the continuous counterparts as $\lambda \to 0$, while the discrete Bellman equation and the Lyapunov design are developed without a convergence claim. The paper works out monitored-qubit examples, implements the dynamic-programming recursion numerically on an invariant circle of density matrices (where the optimal feedback functions exhibit discontinuities), and constructs an explicit Lyapunov feedback law. Crucially, the convergence claims are asserted informally: the rigorous limit theorems the paper cites for them (Lindsay–Parthasarathy 1988, Attal–Pautrat 2006, Gough 2004, Gough–Sobolev 2004) concern uncontrolled models or uncontrolled filters, so the controlled case was a genuine gap from the outset.

Two strengths of convergence must be distinguished. Strong convergence means the discrete unitary (or filter) processes converge in the operator topology of the limiting Fock-space dilation — the quantum analogue of pathwise convergence — while weak convergence means only convergence in law of the trajectories of system observables. The Trotter–Kato-type theorems that close the uncontrolled case yield strong convergence; the available controlled results yield convergence in law of trajectories via the Kurtz–Protter semimartingale limit theorem. Under separated feedback the discrete interaction unitaries depend on the realized measurement record, so the deterministic semigroup-decomposition machinery of the uncontrolled theory does not directly apply — this is the technical heart of the problem.

Problem Statement

Fix a separable system Hilbert space (arbitrary dimension, possibly infinite, not only a two-level system), a horizon $T > 0$, and the repeated-interaction discretization with time step $\lambda^2 = T/k$: $k$ independent two-level field slices — and, when the limiting field multiplicity is $d > 1$, $d$ independent chains of such slices interacting with the same system — with slice $l$ interacting with the system through a unitary $U_l(u)$ with the standard $\lambda$-scaling of the repeated-interaction scheme (as defined for the qubit model in Bouten–van Handel–James 2009 and, for the enlarged system class, in the sense of the uncontrolled-limit literature, Attal–Pautrat 2006; Bouten–van Handel 2008), depending on a control $u$ drawn from a compact admissible set. Admissible control strategies are either open-loop sequences or separated (Markovian state-feedback) strategies in which $u$ is a possibly merely measurable function of the current filter state; the filter is the exact nonlinear (Belavkin-type) conditional-state recursion driven by the realized discrete measurement outcomes. The dynamic-programming layer consists of the Bellman value functions obtained by backward recursion over the controlled filter transitions, and of separated policies selecting minimizers of that recursion.

(1) Closed-loop convergence. Prove or disprove: as $\lambda \to 0$, for the full Hudson–Parthasarathy coefficient class in the limit — matching the generality established in the uncontrolled case (Bouten–van Handel 2008): arbitrary system operators $(S, L, H)$, arbitrary field multiplicity, and unbounded limit coefficients as arise for linear quantum systems — the closed-loop controlled dynamics and the conditional-state (filter) processes of the discrete model converge to the solution of the corresponding controlled quantum stochastic differential equation and the controlled nonlinear filtering equation, in a topology at least as strong as convergence in law of trajectories on path space; and determine whether strong convergence (in the operator topology of the Fock-space dilation) holds when the control is allowed to depend on the measurement record, or whether only weaker topologies are attainable in general.

(2) Policy convergence. Prove or disprove: the discrete Bellman value functions converge, e.g. locally uniformly on the state space of density matrices or in the viscosity-solution sense, to the value function of the continuous-time partially observed control problem, and optimal or ε-optimal separated policies from the discrete recursions converge (possibly along subsequences, in an explicitly specified sense) to optimal controls of the continuous-time problem.

The problem is solved by any one of: (i) a complete positive theory for the full class above; (ii) a counterexample within the class — a concrete model and admissible separated-strategy class for which the closed-loop filter trajectories or the Bellman value functions provably fail to converge as $\lambda \to 0$, or converge to a limit distinct from the natural continuous-time object; or (iii) a rigorous theory for a precisely delimited subclass (e.g. bounded coefficients, specified measurement channels, and a feedback-law class — possibly merely measurable — rich enough to include the discontinuous optimal feedback laws of the source paper's worked examples) that clearly states which features of measurement-based feedback obstruct or permit convergence and covers the source paper's worked examples. Route (i) must establish both statements (1) and (2) for the full class; route (ii) is satisfied by a counterexample to either statement; and route (iii) must establish both statements (1) and (2), restricted to the delimited subclass.

References

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