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The method of complementary channel for analysis of reversibility(sufficiency) of a quantum channel with respect to families of input states(pure states for the most part) are considered and applied to Bosonic linear(quasi-free) channels, in particular, to Bosonic Gaussian channels.  The obtained reversibility conditions for Bosonic linear channels have clearphysical interpretation and their sufficiency is also shown by explicitconstruction of reversing channels. The method of complementary channel givespossibility to prove necessity of these conditions and to describe all reversedfamilies of pure states in the Schrodinger representation.  Some applications in quantum information theory are considered.  Conditions for existence of discrete classical-quantum subchannels and ofcompletely depolarizing subchannels of a Bosonic linear channel are obtained inthe Appendix.

Reversibility of a quantum channel: general conditions and their applications to Bosonic linear channels