  We define a quantity, the so-called purity fidelity, which measures the rate
of dynamical irreversibility due to decoherence, observed e.g in echo
experiments, in the presence of an arbitrary small perturbation of the total
(system + environment) Hamiltonian. We derive a linear response formula for the
purity fidelity in terms of integrated time correlation functions of the
perturbation. Our relation predicts, similarly to the case of fidelity decay,
faster decay of purity fidelity the slower decay of time correlations is. In
particular, we find exponential decay in quantum mixing regime and faster,
initially quadratic and later typically gaussian decay in the regime of
non-ergodic, e.g. integrable quantum dynamics. We illustrate our approach by an
analytical calculation and numerical experiments in the Ising spin 1/2 chain
kicked with tilted homogeneous magnetic field where part of the chain is
interpreted as a system under observation and part as an environment.
