  Intriguing features of the distance between two arbitrary states of an open
quantum system are identified that are induced by initial system-environment
correlations. As an example, we analyze a qubit dephasingly coupled to a
bosonic environment. Within tailored parameter regimes, initial correlations
are shown to substantially increase a distance between two qubit states
evolving to long-time limit states according to exact non-Markovian dynamics.
It exemplifies the breakdown of the distance contractivity of the reduced
dynamics.
