  We introduce two information-theoretical invariants for the projective
unitary group acting on a finite-dimensional complex Hilbert space: PVM- and
POVM-dynamical (quantum) entropies. They quantify the randomness of the
successive quantum measurement results in the case where the evolution of the
system between each two consecutive measurements is described by a given
unitary operator. We study the class of chaotic unitaries, i.e., the ones of
maximal entropy or, equivalently, such that they can be represented by suitably
rescaled complex Hadamard matrices in some orthonormal bases. We provide
necessary conditions for a unitary operator to be chaotic, which become also
sufficient for qubits and qutrits. These conditions are expressed in terms of
the relation between the trace and the determinant of the operator. We also
compute the volume of the set of chaotic unitaries in dimensions two and three,
and the average PVM-dynamical entropy over the unitary group in dimension two.
We prove that this mean value behaves as the logarithm of the dimension of the
Hilbert space, which implies that the probability that the dynamical entropy of
a unitary is almost as large as possible approaches unity as the dimension
tends to infinity.
