Exploring Quantum Chaos through AFL Entropy: Insights from Perturbed Quantum Cat Maps

Dynamical entropies characterize chaos within a dynamical system. One such candidate for quantum dynamical systems is the Alicki-Fannes-Lindblad (AFL) entropy, which is closely related to the entropy exchange of quantum measurement. AFL entropy has been used to study certain quantum systems with a chaotic classical limit, where it recovers the classical Kolmogorov-Sinai entropy. However, some of these systems are not quantum chaotic, and the question arises as to whether quantum dynamical entropies can be used as a diagnosis for quantum chaos. To address this problem, we compute AFL entropy on the perturbed quantum cat maps, a class of single-particle quantum mechanical models that undergo transition between quantum chaotic and non-chaotic regimes when the perturbation strength is tuned. We compare the behaviors of AFL entropy in the early time between the two regimes. We also generalize our study to many-body systems such as the mixed-field Ising models. Furthermore, we discuss the relation and distinction between AFL entropy and other candidates of quantum dynamical entropies, in particular the Connes-Størmer entropy, and demonstrate it numerically in several models.