Quantum Markov State Models for Metastable Dynamics

Abstract

Open quantum systems can rapidly lose most microscopic information, leaving only a few degrees of freedom to govern their long-time dynamics. Classical Markov state models (MSMs) describe such metastable dynamics as transitions among a few representative phases and are widely used to reduce complex-system dynamics in condensed matter and chemical physics. In quantum systems, however, phase labels alone are insufficient when coherence persists between metastable states. Even when the slow modes are known, their spectral projection need not produce valid quantum states. We construct quantum Markov state models (QMSMs) that describe the surviving information and its evolution on a small physical state space containing classical sectors and quantum matrix blocks. We quantify metastability by assuming that the evolution channel $\mathcal C$ changes little when applied a second time, with sufficiently small defect $η=\|\mathcal C^2-\mathcal C\|_\diamond$, and that the number of slow degrees of freedom is bounded independently of the full system size. Under these assumptions, we construct compression and reconstruction channels whose composition recovers every reduced state exactly, while the reverse composition gives an exactly idempotent channel approximating $\mathcal C$, answering Kitaev's exact-rounding question [Kit25]. Our constructions give an optimal diamond-norm bound of $\mathcal{O}(η^{1/3})$ on all input states, as well as an improved bound of $\mathcal{O}(η^{1/2})$ on metastable states prepared by $\mathcal C$, with constants depending only on the slow dimension. The reduced transition channel can be iterated to predict the microscopic dynamics with controlled error. We illustrate the QMSM through a weakly driven dissipative spin chain supporting either a metastable logical qubit or long-lived classical phases.