Bell-type inequalities of $$l_1$$ l 1 -norm of coherence

In this paper, we investigate the Bell-type inequalities of $$l_1$$ -norm of coherence for two- or three-qubit system and establish the relation between the $$l_1$$ -norm of coherence and the degree of coherence. It is found that the upper bounds of the inequalities can be much stronger. The Bell-type inequality of $$l_1$$ -norm of coherence for two-qubit system can be violated by all two-qubit entangled pure states, and thus, it can serve as an entanglement witness. The Bell-type inequalities can also be violated by quantum systems which pass through the uncorrelated or correlated channels. Particularly, the partial correlations among consecutive uses of the correlated channel can enhance the value of the left-hand side of the Bell-type inequalities.