Symmetry Properties of Quantum Dynamical Entropy

Chaos in classical dynamical systems can be quantified by the Kolmogorov-Sinai (KS) entropy. KS entropy has been generalized to quantum systems as the Alicki-Fannes-Lindblad (AFL) entropy to measure chaos in the semiclassical limit and in general infinite-dimensional quantum systems. However, the story is more subtle in finite dimensions, as (1) there are systems which are not quantum chaotic but nonetheless have a chaotic classical limit and (2) the sensitvity of AFL entropy to symmetries and the classical limit is basis-dependent. We study this basis-dependence of AFL entropy analytically and numerically to determine its applicability for diagnosing finite-dimensional quantum chaos, using quantum maps as demonstrative examples. We also comment on the relation to correlation function decays as given by Ruelle-Pollicott resonances.