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 $p\in (1,\infty )$.

• Received 10 November 2015

DOI:https://doi.org/10.1103/PhysRevA.93.012110