Quantum model reduction for continuous-time quantum filters

TL;DR

This work addresses the challenge of simulating and controlling open quantum systems by reducing stochastic quantum filters while exactly preserving the output statistics of a chosen set of observables. It develops a reduction framework based on the non-observable subspace and non-commutative conditional expectations to yield reduced filters that remain in Belavkin form. The method combines Krylov-type observability ideas, Wedderburn decompositions, and CPTP factorization to produce minimal algebras and reduced generators that reproduce the target observables with $\Theta_t^j = \Xi_t^j$ for all $t$ and $j$. Demonstrations on generalized quantum non-demolition scenarios and spin-chain models show exact reproduction of observables and offer scalable sub-optimal reductions for larger systems, enabling efficient quantum trajectory simulation and feedback design.

Abstract

The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter's state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task.In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability.This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.

Figures (4)

• Figure 1: Schematic of the use of the original filter $\Sigma$ and reduced filters ${\Sigma_L}$ and $\Sigma_Q$.
• Figure 2: Graphical representation of the reduced model when the algebra $\mathscr{A}$ is or is not $\mathcal{L}^*$-, $\mathcal{G}_{D_j}^*$- and $\mathcal{K}_{C_j}^*$-invariant.
• Figure 3: Numerical simulations for the measured spin chain with $N=4$, $\delta_j$ sampled from a Gaussian distribution with mean $2$ and standard deviation $0.2$, $\mu_j$ sampled from a Gaussian distribution with mean $1$ and standard deviation $0.2$, and $\gamma_j =\gamma$ and $\alpha_j=\alpha$ for all $j$. The left column shows a diffusive-type evolution, i.e. $\gamma=0.5$, $\alpha=0$, while the right column shows a counting-type evolution, i.e. $\gamma=0$, $\alpha=4$. From top to bottom we have: Comparison of the population in the standard basis versus time $\left< O_j(t) \right>$ for the original (dots) and reduced (continuous curves) filters; Comparison of the local magnetization versus time $\left< \sigma_z^{(j)}(t) \right>$ for the original (empty circles) and reduced (continuous curves) filters; Difference between the fidelity between a filter initialized in the correct initial condition $\rho_0$ and a filter initialized in a random initial condition $\rho_0^e$ for the original and reduced model.
• Figure 4: Pictorial representation of the block-diagonal index $e\in[-N+1,N-1]$ of the principal block diagonals.

Theorems & Definitions (40)

• Definition 1: Indistinguishable states and non-observable subpace.
• Lemma 1
• proof
• Proposition 1
• proof
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2: prxq2024