  Using the symplectic tomography map, both for the probability distributions
in classical phase space and for the Wigner functions of its quantum
counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.
Because the marginal distributions, obtained by the tomography map, are always
well defined probabilities, the correspondence between classical and quantum
notions is very clear. Then we also obtain the corresponding expressions in
Hilbert space. Some examples are worked out. Classical and quantum exponents
are seen to coincide for local and non-local time-dependent quadratic
potentials. For non-quadratic potentials classical and quantum exponents are
different and some insight is obtained on the taming effect of quantum
mechanics on classical chaos. A detailed analysis is made for the standard map.
Providing an unambiguous extension of the notion of Lyapunov exponent to
quantum mechnics, the method that is developed is also computationally
efficient in obtaining analytical results for the Lyapunov exponent, both
classical and quantum.
