Stochastic limits of quantum repeated measurements with memory: higher dimensions, general memory scalings, and uniform error control

In repeated-interaction (collision) models of open quantum systems, a small system collides sequentially with identical environment units and is projectively measured after each collision, producing a discrete-time random evolution of its conditional state (a quantum trajectory). Jacquier, Kardaras and Viot (arXiv:2512.11462, 2025) proved, for a two-level system with small fixed environment dimension only, that under the short-interaction scaling $\tau=1/n$ together with a memory parameter $p(n)=\mathrm{e}^{-\Gamma/n}$, the discrete dynamics converge to Volterra equations with exponential convolution kernel: a deterministic Volterra integro-differential equation for a measurement-free stochastic swap of adjacent environment units, and, in the measurement model with a probabilistic reset-type memory mechanism, a stochastic Volterra SDE; and that unitary couplings perturbed by quantum channels obey, at the level of the continuous limits, an asymptotic $O(\varepsilon)$ deviation bound in expectation and a small-noise large deviations principle. The open problem, arising from the scope restrictions of these results and from the authors' own remark that the work 'sets interesting future research paths at the intersection of quantum algorithms, stochastic differential equations, weak convergence and large deviations', is to extend this rigorous discrete-to-continuous limit theory to the generality needed for it to serve as a certified approximation of the finite discrete models: (i) arbitrary finite-dimensional systems and environments and observables with arbitrarily many outcomes; (ii) memory scalings $p(n)$ other than $\mathrm{e}^{-\Gamma/n}$ and other memory mechanisms, determining which continuous-time limits (if any) they produce; and (iii) quantitative approximation-error and large-deviation control uniform in the number of steps $n$ and the perturbation strength $\varepsilon$ at finite parameters.

Background

A repeated-interaction or collision model describes a small quantum system with Hilbert space $\mathcal{H}_0$ interacting one after another with an infinite chain of identical environment units, each with Hilbert space $\mathcal{H}$ (Attal and Pautrat 2006; Ciccarello et al. 2022). Each collision lasts a short time $\tau$ and is governed by a unitary operator $U=\mathrm{e}^{-\mathrm{i}\tau H_{\mathrm{tot}}}$; immediately afterwards a fixed observable $A$ of the just-collided environment unit is measured projectively, and the outcome updates the conditional state of the system. The resulting discrete-time random sequence of conditional states is called a quantum trajectory.

In continuous time the corresponding object is the Belavkin equation, also called a stochastic Schrödinger equation: a stochastic differential equation (SDE) for the state of an open quantum system under continuous indirect measurement (Belavkin 1994). In the diffusive case the state $\rho_t$, a density operator (positive Hermitian matrix of unit trace), solves an SDE with drift of Lindblad type, $L(\rho)=C\rho C^\dagger-\mathrm{i}[H_0,\rho]-\tfrac{1}{2}\{CC^\dagger,\rho\}$ where $C$ is a coupling operator, $H_0$ a Hamiltonian and $\{X,Y\}=XY+YX$ the anticommutator, and with diffusion coefficient $\Theta_C(\rho)=C\rho+\rho C^\dagger-\mathrm{Tr}[\rho(C+C^\dagger)]\rho$ driven by a Brownian motion.

Pellegrini (2008) proved the foundational convergence result connecting the two pictures: for a two-level system ($\mathcal{H}_0=\mathcal{H}=\mathbb{C}^2$) and a two-outcome observable, under the short-interaction scaling $\tau=1/n$, the discrete quantum trajectories converge to the solution of the diffusive Belavkin equation; the jump-type case for two-level systems followed in 2010. Related limits in other formulations were obtained by Gough and Sobolev (2004) for discrete filtering and by Attal and Joye (2007) in a weak-coupling regime.

Jacquier, Kardaras and Viot (2025) extended this program, still for a two-level system with small fixed environment dimension (the environment Hilbert space is $\mathbb{C}^2$ in the convergence and alternating-coupling sections and $\mathbb{C}^8$ in the channel-robustness section, where the measured observable acts on $\mathbb{C}^4$ and has four outcomes). First, couplings alternating between two unitaries according to the parity of the step. Second, robustness to noisy implementation: the unitary is replaced by a quantum channel, i.e. a completely positive trace-preserving map on density operators expressible in Kraus form $\mathcal{E}(\rho)=\sum_k K_k\rho K_k^\dagger$, and, at the level of the continuous limits, the perturbed limiting trajectories deviate from the ideal ones by an $O(\varepsilon)$ bound in expectation (their Proposition 3.24, $\varepsilon$ the channel strength; their Theorem 3.25 strengthens this to convergence in distribution to zero of the $\varepsilon^\alpha$-rescaled deviation for $\alpha<1/2$) and satisfy a large deviations principle (their Theorem 3.26); a large deviations principle (LDP) means that deviation probabilities decay exponentially with a computable rate function, obtained here through the McKean-Vlasov Freidlin-Wentzell LDP of dos Reis, Salkeld and Tugaut (2019). Third, the main novelty, memory with parameter $p(n)=\mathrm{e}^{-\Gamma/n}$, analyzed for two distinct mechanisms. In the first (their Section 4.1), before each unitary step a stochastic swap acts on adjacent environment units with probability $p(n)$, so that the reduced system dynamics retains information about past states; in the measurement-free setting the rescaled reduced states are proved to converge to a deterministic Volterra integro-differential equation (their Theorem 4.2). In the second (their Section 4.2), a probabilistic reset mechanism: with probability $1-p(n)$ the post-measurement state returns to the initial global state; in the measurement model the rescaled conditional states converge to the stochastic Volterra SDE $\rho_t=\rho_0+\int_0^t\mathrm{e}^{-\Gamma(t-s)}L(\rho_s)\,\mathrm{d}s+\int_0^t\mathrm{e}^{-\Gamma(t-s)}\Theta_C(\rho_s)\,\mathrm{d}W_s$ (their Theorem 4.8), whose exponential convolution kernel $\mathrm{e}^{-\Gamma(t-s)}$ arises specifically from the exponential form of $p(n)$.

Memory in repeated-interaction models has also been studied on the physics side, without rigorous continuous-time limit theorems: Pellegrini and Petruccione (2009) derived discrete non-Markovian master equations by projection-operator techniques; Barchielli, Pellegrini and Petruccione (2012) built non-Markovian stochastic Schrödinger equations by randomizing coefficients of the linear equation; Pleasance, Neira, Merkli and Petruccione (2025) analyzed ancilla-ancilla entanglement as a memory mechanism, mapping it onto an enlarged Markovian collision model. The problem below asks for the rigorous, quantitatively controlled limit theory for such memory-carrying repeated measurement models beyond the regime in which it is currently proved.

Problem Statement

The problem is to extend the rigorous discrete-to-continuous stochastic limit theory for quantum repeated measurement models with environment memory, established by Jacquier, Kardaras and Viot (2025) for two-level system and environment under the scaling $\tau=1/n$ and $p(n)=\mathrm{e}^{-\Gamma/n}$ (a deterministic Volterra limit for the measurement-free swap memory and, in the measurement model, a stochastic Volterra limit for the reset-type memory), to the generality needed for the continuous limits to serve as certified approximations of the finite discrete models. Consider the repeated-interaction model in which a system with Hilbert space $\mathcal{H}_0$ collides sequentially with environment units with Hilbert space $\mathcal{H}$ under a unitary coupling (possibly alternating between steps, or perturbed by a quantum channel of strength $\varepsilon$), each collision being followed by a projective measurement of a fixed observable on the colliding unit, together with a memory mechanism with probability parameter $p(n)$, such as random adjacent-environment swaps or probabilistic resetting toward the initial state. The following linked questions are open:

(i) Dimension generality. Determine whether, and in which limiting equations, the discrete quantum trajectories converge, with explicit small-step expansions, for arbitrary finite-dimensional $\mathcal{H}_0$ and $\mathcal{H}$ and observables with arbitrarily many (finitely many) outcomes; the present proofs and expansions are specific to two-level systems with small fixed environment dimension ($\mathcal{H}_0=\mathcal{H}=\mathbb{C}^2$ in the convergence and alternating-coupling sections, $\mathcal{H}=\mathbb{C}^8$ in the channel-robustness section) and to few-outcome observables (two outcomes in the convergence and alternating-coupling sections; a single four-outcome observable on $\mathbb{C}^4$ with a qubit system in the channel-robustness section), and, for the swap memory combined with projective measurements, only the measurement-free deterministic limit is proved, so the stochastic convergence with swap memory is open even at the two-level scope.

(ii) Memory scaling and mechanism classification. Determine the continuous-time limit, if any, over the whole class of memory parameters $p(n)$ other than $\mathrm{e}^{-\Gamma/n}$ (illustrated by decay of the memory deviation $1-p(n)$ faster or slower than order $1/n$), and over the whole class of other memory mechanisms proposed in the literature (illustrated by stochastic swaps between environment units that are not restricted to adjacent units or follow a general probability law over pairs, and initial intra-environment correlations or entanglement in the sense of Pleasance, Neira, Merkli and Petruccione (2025)); for this whole class, either identify the resulting class of convolution kernels or non-Volterra dynamics, or exhibit the class of scalings and mechanisms for which no continuous-time limit exists. The known exponential kernel arises specifically from the exponential form of $p(n)$.

(iii) Uniform error control. Establish quantitative approximation-error bounds and large-deviation-based control, uniform in the number of discrete steps $n$ and the channel-perturbation strength $\varepsilon$ and valid at finite $n$ and $\varepsilon$, for the discrete models relative to their continuous limits; the existing robustness estimates compare the perturbed and the ideal continuous limiting trajectories with one another and are asymptotic as $\varepsilon\to0$ only (the $O(\varepsilon)$ expectation bound of Proposition 3.24, the $\varepsilon^\alpha$-rescaled convergence in distribution of Theorem 3.25, and the small-noise LDP of Theorem 3.26), while the discrete-to-continuous convergences carry no rates uniform in $n$.

The common objective of (i)-(iii) is one certified continuous-time limit theory for memory-carrying repeated quantum measurements; the facets are restriction axes of the same convergence program, each to be proved rigorously for the stated model, and all three are jointly required: a rigorous resolution of any single facet alone does not resolve the task. The source paper additionally motivates applications of the limiting SDEs to quantum-algorithm analysis and noise mitigation on NISQ (noisy intermediate-scale quantum) hardware; that calibration-and-validation program has no recognizable completion boundary and is motivation for, not part of, this problem.

References

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