  A set of schemes for secure quantum communication are analyzed under the
influence of non-Markovian channels. By comparing with the corresponding
Markovian cases, it is seen that the average fidelity in all these schemes can
be maintained for relatively longer periods of time. The effects of
non-Markovian noise on a number of facets of quantum cryptography, such as
quantum secure direct communication, deterministic secure quantum communication
and their controlled counterparts, quantum dialogue, quantum key distribution,
quantum key agreement, etc., have been extensively investigated. Specifically,
a scheme for controlled quantum dialogue (CQD) is analyzed over damping,
dephasing and depolarizing non-Markovian channels, and subsequently, the effect
of these non-Markovian channels on the other schemes of secure quantum
communication is deduced from the results obtained for CQD. The damped
non-Markovian channel causes, a periodic revival in the fidelity; while
fidelity is observed to be sustained under the influence of the dephasing
non-Markovian channel. The depolarizing channel, as well as the other
non-Markovian channels discussed here, show that the obtained average fidelity
subjected to noisy environment depends on the strength of coupling between the
quantum system with its surroundings and the number of rounds of quantum
communication involved in a particular scheme.
