Notes on quantum coherence with l_1-norm and convex-roof l_-norm

Abstract: In this work, we evaluate quantum coherence using the l_1-norm and convex-roof l_1-norm and obtain several new results. First, we provide some new general triangle-like inequalities of quantum coherence, with results better than existing ones. Second, for some special three-dimensional quantum states, a method for calculating the convex-roof l_1-norm is presented. Lastly, we offer distinct upper bounds in the l_1-norm measure of coherence based on the quantum state itself.