Half-delocalization for semiclassical measures of split higher-dimensional quantum cat maps

We study quantum cat maps on multidimensional tori and prove an optimal lower bound for the Kolmogorov-Sinai entropy of their semiclassical measures under the hypothesis that the dynamics splits over the rationals. This is the first instance of such a bound for a dynamical system with multiple positive Lyapunov exponents. The proof is based on a new notion of conditional quantum dynamical entropy and a careful study of the multiple Ehrenfest time adapted to the dynamics.