  Entanglement between a quantum system and its environment leads to loss of
coherence in the former. In general, the temporal fate of coherences is
complicated. Here, we establish the connection between decoherence of a central
system and fidelity decay in the environment for a variety of situations,
including both, energy conserving and dissipative couplings. We show how
properties of unitary time evolution of the environment can be inferred from
the non-unitary evolution of coherences in the central system. This opens up
promising ways for measuring Loschmidt echoes in a variety of situations.
