  We propose to study the $L^2$-norm distance between classical and quantum
phase space distributions, where for the latter we choose the Wigner function,
as a global phase space indicator of quantum-classical correspondence. For
example, this quantity should provide a key to understand the correspondence
between quantum and classical Loschmidt echoes. We concentrate on fully chaotic
systems with compact (finite) classical phase space. By means of numerical
simulations and heuristic arguments we find that the quantum-classical fidelity
stays at one up to Ehrenfest-type time scale, which is proportional to the
logarithm of effective Planck constant, and decays exponentially with a maximal
classical Lyapunov exponent, after that time.
