Model Reduction for Controlled Quantum Markov Dynamics

TL;DR

This work extends observable-based model reduction from time-independent to time-dependent Lindblad dynamics under control, enabling exact replication of selected observable trajectories with a smaller quantum system. By constructing the Krylov observable subspace $\\mathscr{O}$ and its unital algebra $\\mathscr{A}$, the authors define a CPTP projection that produces a Lindblad-form reduced generator $\\check{\\mathcal{L}}_u$ and reduced observables $\\check{O}$, guaranteeing physically valid reduced dynamics. Two practical sufficient conditions are provided to certify reducibility without full algebra construction, facilitating scalable application. The approach is demonstrated on a controlled dephasing central-spin model, where the bath reduces to a classical ensemble and the reduced dynamics decouple into independent blocks, allowing efficient computation of target observables. The framework promises broader applicability to Floquet-Lindblad systems, switching controls, and reachable MR in open quantum systems, potentially enabling real-time filtering and control with significantly lower dimensional models.

Abstract

We consider the problem of model reduction for Markovian quantum systems whose dynamics are described by a time-dependent Lindblad generator -- notably, as arising in the presence of external control. Our approach, which builds upon Krylov operator subspaces and operator-algebraic techniques introduced for time-independent generators, returns a reduced model that reproduces exactly the evolution of observables of interest and is guaranteed to be in Lindblad form.

Theorems & Definitions (9)

• Proposition 1
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• Proposition 2
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• Theorem 1
• Proposition 3
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• Proposition 4
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