  Decoherence is all around us. Every quantum system that interacts with the
environment is doomed to decohere. The preservation of quantum coherence is one
of the major challenges faced in quantum technologies, but its use as a
resource is very promising and can lead to various operational advantages, for
example in quantum algorithms. Hence, much work has been devoted in recent
years to quantify the coherence present in a system. In the present paper, we
formulate the quantum resource theory of dynamical coherence. The underlying
physical principle we follow is that the free dynamical objects are those that
cannot preserve or distribute coherence. This leads us to identify classical
channels as the free elements in this theory. Consequently, even the quantum
identity channel is not free as all physical systems undergo decoherence and
hence, the preservation of coherence should be considered a resource. In our
work, we introduce four different types of free superchannels (analogous to
MIO, DIO, IO, and SIO) and discuss in detail two of them, namely,
dephasing-covariant incoherent superchannels (DISC), maximally incoherent
superchannels (MISC). The latter consists of all superchannels that do not
generate non-classical channels from classical ones. We quantify dynamical
coherence using channel-divergence-based monotones for MISC and DISC. We show
that some of these monotones have operational interpretations as the exact, the
approximate, and the liberal coherence cost of a quantum channel. Moreover, we
prove that the liberal asymptotic cost of a channel is equal to a new type of
regularized relative entropy. Finally, we show that the conversion distance
between two channels under MISC and DISC can be computed using a semi-definite
program (SDP).
