  We plot the geometry of several distance-based quantifiers of coherence for
Bell-diagonal states. We find that along with both $l_{1}$ norm and relative
entropy of coherence changes continuously from zero to one, their surfaces move
from the separable regions to the entangled regions. Based on this fact, it is
more illuminating to use an intuitive geometry to explain quantum states with
nonzero coherence can be used for entanglement creation, rather than the other
way around. We find the necessary and sufficient conditions that quantum
discord of Bell-diagonal states equal to its relative entropy of coherence and
depict the surfaces of the equality. We give surfaces of relative entropy of
coherence for $X$ states. We show the surfaces of dynamics of relative entropy
of coherence for Bell-diagonal states under local nondissipative channels and
find that all coherence under local nondissipative channels decrease.

Quantum Physics