Mean Dynamical Entropy of Quantum Maps on the Sphere Diverges in the Semiclassical Limit

We analyze quantum dynamical entropy based on the notion of coherent states. The mean value of this quantity for quantum maps on the sphere is computed as an average over the uniform measure on the space of unitary matrices of size $N$. Mean dynamical entropy is positive for $N\ensuremath{\ge}3$, which supplies a direct link between random matrices of the circular unitary ensemble and the chaotic dynamics of the corresponding classical maps. Mean entropy tends logarithmically to infinity in the semiclassical limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and this indicates the ubiquity of chaos in classical mechanics.