Effect of unital and non-unital channels on faithfulness of two-qubit entanglement

States whose entanglement can be detected by a fidelity-based entanglement witness are faithful. In this paper, we introduce a new necessary and sufficient condition for identifying faithful states. This condition is based on the trace norm. With the help of this condition, we investigate the effect of local channels on the faithfulness of two-qubit entangled states. The result shows that all the unital channels cannot enhance the faithfulness of entangled states. Some non-unital channels can improve the faithfulness of a restricted set of entangled states and even change an unfaithful entangled state to a faithful one. However, the faithfulness of all the states with maximally mixed marginals can only decrease when subjected to local channels.