  We study the stability of unitary quantum dynamics of composite systems (for
example: central system + environment) with respect to weak interaction between
the two parts. Unified theoretical formalism is applied to study different
physical situations: (i) coherence of a forward evolution as measured by purity
of the reduced density matrix, (ii) stability of time evolution with respect to
small coupling between subsystems, and (iii) Loschmidt echo measuring dynamical
irreversibility. Stability has been measured either by fidelity of pure states
of a composite system, or by the so-called reduced fidelity of reduced density
matrices within a subsystem. Rigorous inequality among fidelity,
reduced-fidelity and purity is proved and a linear response theory is developed
expressing these three quantities in terms of time correlation functions of the
generator of interaction. The qualitatively different cases of regular
(integrable) or mixing (chaotic in the classical limit) dynamics in each of the
subsystems are discussed in detail. Theoretical results are demonstrated and
confirmed in a numerical example of two coupled kicked tops.
