  General semiclassical expression for quantum fidelity (Loschmidt echo) of
arbitrary pure and mixed states is derived. It expresses fidelity as an
interference sum of dephasing trajectories weighed by the Wigner function of
the initial state, and does not require that the initial state be localized in
position or momentum. This general dephasing representation is special in that,
counterintuitively, all of fidelity decay is due to dephasing and none due to
the decay of classical overlaps. Surprising accuracy of the approximation is
justified by invoking the shadowing theorem: twice--both for physical
perturbations and for numerical errors. It is shown how the general expression
reduces to the special forms for position and momentum states and for wave
packets localized in position or momentum. The superiority of the general over
the specialized forms is explained and supported by numerical tests for wave
packets, non-local pure states, and for simple and random mixed states. The
tests are done in non-universal regimes in mixed phase space where detailed
features of fidelity are important. Although semiclassically motivated, present
approach is valid for abstract systems with a finite Hilbert basis provided
that the discrete Wigner transform is used. This makes the method applicable,
via a phase space approach, e. g., to problems of quantum computation.
