Trace-distance measure of coherence

We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, X states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between l(1)-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both l(p)-norm- and Schatten-p-norm-based measures violate the (strong) monotonicity for all p is an element of (1, infinity).