  We study the degree to which the coherence of quantum states is affected by
noise. We give the definition of the $n$-th decay rate and investigate the
coherence of Bell-diagonal states under $n$ iterations of channels. We derive
explicit formulas of the $n$-th decay rates based on $l_1$ norm of coherence,
relative entropy of coherence and skew information-based coherence. It is found
that the larger $n$ is, the faster the $n$-th decay rate decreases as the
parameter $p$ of Bell-diagonal states increases. Moreover, for any fixed $n$,
with the increase of $p$, Bell-diagonal states can be completely incoherent
under generalized amplitude damping (GAD) channels, depolarization (DEP)
channels and phase flip (PF) channels, while this is not the case for bit flip
(BF) channels and bit-phase flip (BPF) channels. We also investigate the
geometry of the relative entropy of coherence and skew information-based
coherence of Bell-diagonal states under different channels when the $n$-th
decay rate is one, i.e., the coherence is frozen. It is shown that compared
with BF and BPF channels, when $n$ is large enough, the coherence of
Bell-diagonal states will not be frozen under GAD, DEP and PF channels. For
skew information-based coherence, similar properties of coherence freezing are
found.

Quantum Physics