A Complexity-Based Approach to Quantum Observable Equilibration

Abstract: We investigate the role of a statistical complexity measure to assign equilibration in isolated quantum systems. While unitary dynamics preserve global purity, expectation values of observables often exhibit equilibration-like behavior, raising the question of whether complexity can track this process. In addition to examining observable equilibration, we extend our analysis to study how the complexity of the quantum states evolves, providing insight into the transition from initial coherence to equilibrium. We define a classical statistical complexity measure based on observable entropy and deviation from equilibrium, which captures the dynamical progression towards equilibration and effectively distinguishes between complex and non-complex trajectories. In particular, our measure is sensitive to non-complex dynamics, such as the quasi-periodic behavior exhibited by low effective dimension initial states, where the systems explore a limited region of the Hilbert space as they oscillate in an informational coherence-preserving manner. These findings are supported by numerical simulations of an Ising-like non-integrable Hamiltonian spin-chain model. Our work provides new insight into the emergence of equilibrium behavior from unitary dynamics and advances complexity as a meaningful tool in the study of the emergence of classicality in microscopic systems.