  We study partial coherence and its connections with entanglement. First, we
provide a sufficient and necessary condition for bipartite pure state
transformation under partial incoherent operations: A bipartite pure state can
be transformed to another one if and only if a majorization relationship holds
between their partial coherence vectors. As a consequence, we introduce the
concept of maximal partial coherent states in the sense that they can be used
to construct any bipartite state of the same system via partial incoherent
operations. Second, we provide a strategy to construct measures of partial
coherence by the use of symmetric concave functions. Third, we establish some
relationships between partial coherence and entanglement. We show that the
minimal partial coherence under local unitary operations is a measure of
entanglement for bipartite pure states, which can be extended to all mixed
states by convex-roof. We also show that partial coherence measures are induced
through maximal entanglement under partial incoherent operations for bipartite
pure states. There is a one-to-one correspondence between entanglement and
partial coherence measures.
