Symmetry properties of quantum dynamical entropy

As quantum analogs of the classical Kolmogorov-Sinai entropy, quantum dynamical entropies have emerged as important tools to characterize complex quantum dynamics. In particular, Alicki-Fannes-Lindblad (AFL) entropy, which quantifies the information production rate of a coherent quantum system subjected to repeated measurement, has received considerable attention as a potential diagnostic for quantum chaos. Despite this interest, the precise behavior of quantum dynamical entropy in the presence of symmetry remains largely unexplored. In this work, we establish rigorous inequalities of the AFL entropy for arbitrary unitary dynamics (single particle and many body) in the presence of various types of symmetry. Our theorems encompass three cases: Abelian symmetry, an anticommuting unitary, and non-Abelian symmetries. In particular, we show that, while the cumulative AFL entropy generally saturates to the dimensional bound at late times for chaotic dynamics, this saturation value is distinctively lower when the measurements respect the symmetries. We motivate our main results with numerical simulations of the perturbed quantum cat maps. Our findings highlight the crucial role of symmetry in quantum dynamics under measurements, and our framework is readily adaptable for investigating symmetry's influence across diverse probes of quantum chaos.