  Bell inequality is a mathematical inequality derived using the assumptions of
locality and realism. Its violation guarantees the existence of quantum
correlations in a quantum state. Bell inequality acts as an entanglement
witness in the sense that a pure bipartite quantum state, having nonvanishing
entanglement, always violates a Bell inequality. We construct Bell-type
inequalities for product states in quantum coherence theory for different
measures of coherence, and find that the maximally entangled states violate
these inequalities. We further show that Bell-type inequalities for relative
entropy of coherence is violated by all two-qubit pure entangled states,
serving as an entanglement witness.
