Rigorous continuous-time limit and error control for collision-model quantum filtering with superposed-coherent input states

Dąbrowska (Quantum Information Processing 18, 224 (2019)) derived stochastic master (filtering) equations, and hence quantum trajectories, for a finite-dimensional quantum system driven by a Bose field prepared in a superposition of two continuous-mode coherent states, using a collision model: an infinite chain of non-interacting qubits, prepared in an entangled discrete analogue of the superposed-coherent field state, that interact with the system one at a time and are then measured (photon counting or homodyne detection). The derivation is formal at three points: expansions are truncated at linear order in the collision duration τ; the discrete 'toy Fock' qubit-chain surrogate of the field is used without any approximation-error bound; and the τ→0 limit is taken by discarding terms 'that do not contribute', with no convergence notion, no rate, and no existence/uniqueness statement. The open problem is to supply the missing mathematical foundations — an explicit error bound for the discrete state approximation, control of the neglected higher-order τ corrections, and a proof of existence, uniqueness, and strong convergence with rates of the discrete conditional dynamics to the limiting coupled stochastic master equations for both measurement schemes. The rigorous convergence theorems available for repeated-interaction models cover only product-state (Gaussian-input) probe chains, and all subsequent work in this line (single-photon, N-photon, two-photon, squeezed-number-state inputs) retains the formal character.

Background

An open quantum system interacting with a quantized radiation (Bose) field can be monitored continuously in time by detecting the outgoing field, e.g. by photon counting or by homodyne detection (a scheme in which a field quadrature is measured by interference with an intense reference beam). Conditioned on the measurement record, the state of the system evolves stochastically according to a stochastic master equation (SME), also called a quantum filtering equation, whose random solutions are known as quantum trajectories. When the input field is in a Gaussian state (vacuum, coherent, thermal, or squeezed), the input-output theory built on quantum stochastic calculus (QSC) of the Itô type — a quantum analogue of classical stochastic calculus with creation and annihilation noise increments — yields Markovian SMEs, and the corresponding filtering theory is well developed, including rigorous convergence results for discrete approximations: it is known that repeated-interaction ('collision') models, in which the field is replaced by a chain of probes each interacting with the system once for a short time τ and then being measured, converge as τ→0 to the continuous-time quantum stochastic differential equations and filters.

These standard tools break down when the input field is prepared in a non-classical (non-Gaussian) state. A continuous-mode coherent state $|\alpha\rangle$ is a state of the propagating field with a definite complex amplitude envelope $\alpha(t)$; a superposition $c_\alpha|\alpha\rangle+c_\beta|\beta\rangle$ of two such states (a 'cat' state of the traveling field) carries temporal correlations, so the system becomes entangled with the part of the field that has not yet interacted with it, and the conditional dynamics is non-Markovian — described by a coupled set of sector equations rather than a single SME. Two routes around this difficulty exist. The first is a non-Markovian embedding, which adjoins an auxiliary quantum system (an ancilla) that emits the non-classical signal, restoring Markovian filtering on a larger composite. The second is a direct collision model, introduced for superposed-coherent inputs by Dąbrowska (Quantum Information Processing 18, 224 (2019)): the field is modeled by an infinite chain of non-interacting two-level probes (qubits) prepared in the entangled state $c_\alpha\otimes_k|\alpha_k\rangle+c_\beta\otimes_k|\beta_k\rangle$, a discrete analogue of the superposed-coherent field state living on the 'toy Fock space' used for discrete approximations of the symmetric Fock space of QSC; each probe interacts with the system once, during a time interval of length τ through a unitary whose coupling is scaled as $1/\sqrt{\tau}$, and is measured right after its interaction. With a factorized initial system–field state, the paper derives discrete recurrence equations for the conditional state, decomposed into four sectors (αα, αβ, βα, ββ) whose mixing is weighted by products of probe-state overlaps $\langle\beta_k|\alpha_k\rangle$, and then formally passes to τ→0, obtaining coupled Itô SMEs driven by the counting-process innovation $dn_t-\nu_t dt$ (detected-count increment minus its conditional intensity) or the homodyne innovation $dq_t-\mu_t dt$ (quadrature readout minus its conditional mean), consistent with the earlier embedding-based results.

The derivation is, however, formal at exactly the points that a mathematician must control, and these are the origin of the open problem. First, all operator expansions are truncated at linear order in τ, with $O(\tau^2)$ and $O(\tau^{3/2})$ remainders discarded, and their effect on the limit is never assessed. Second, the discrete entangled qubit-chain state is used as a surrogate for the true continuous-mode superposed-coherent field state, but the approximation error, and its scaling with τ and with the mode envelopes, is not quantified. Third, the τ→0 limit is taken by simply dropping terms 'that do not contribute', with no specification of a convergence notion, no existence or uniqueness statement for the limiting coupled SMEs, and no convergence rate. The rigorous convergence theorems available for repeated-interaction models (Gough–Sobolev 2004; Attal–Pautrat 2006; Bouten–van Handel 2008; Bouten–van Handel–James 2009; Attal–Deschamps–Pellegrini 2015) cover chains whose probes are in a product state — vacuum or Gaussian inputs — and therefore do not apply to the entangled input chain that encodes the superposition of coherent states.

Problem Statement

Place the collision-model derivation of quantum filtering equations for an open quantum system driven by a superposition of continuous-mode coherent states on rigorous mathematical footing. Concretely, for the model of Dąbrowska (Quantum Information Processing 18, 224 (2019)) — an infinite chain of non-interacting qubit probes, initial chain state the entangled discrete analogue of $c_\alpha|\alpha\rangle+c_\beta|\beta\rangle$, one interaction of duration τ per probe with coupling scaled as $1/\sqrt{\tau}$, and measurement of each probe after its interaction by photon counting or homodyne detection — establish the following.

1. Approximation error: an explicit bound on the error of the discrete qubit-chain approximation as a surrogate for the true continuous-mode superposed-coherent field state and dynamics, as a function of the probe duration τ, the coherent envelopes $\alpha(t)$ and $\beta(t)$, and the probe-state overlaps $\langle\beta_k|\alpha_k\rangle$ whose products weight the cross-sector initial conditions; and the parameter regime in which the collision model faithfully reproduces the continuous-field dynamics.

2. Higher-order corrections: an analysis of the effect of the neglected $O(\tau^2)$ and $O(\tau^{3/2})$ terms on the discrete recurrence equations and on their τ→0 limit — in particular, whether the leading-order continuous-time filtering equations are correct to the stated order, and whether any higher-order contribution accumulates at fixed physical time $t=j\tau$ as the number of collisions grows.

3. Rigorous limit: a proof of existence and uniqueness of solutions to the limiting coupled Itô stochastic master equations for both the counting and the homodyne measurement schemes, and of strong convergence (convergence of the random conditional states themselves, in an appropriate norm or probabilistic sense) of the discrete conditional dynamics to these solutions as τ→0 with $j\tau=t$ fixed, with an explicit convergence rate.

Deliverables 1–3 form the mathematical core of the problem and each is independently required. In particular, the recurrence-level analysis of the neglected $O(\tau^2)$ and $O(\tau^{3/2})$ terms asked for under item 2 must be carried out explicitly on the discrete recurrence equations, and it is not considered discharged by the convergence proof of item 3, although the two must be mutually consistent. Their common objective is to convert the collision-model route to non-Markovian quantum filtering from a formal derivation into a controlled approximation scheme with a mathematically justified continuous-time limit.

References

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