Ordering states with various coherence measures. (arXiv:1803.02688v1 [quant-ph])
Quantum coherence is one of the most significant theories in quantum physics.
Ordering states with various coherence measures is an intriguing task in
quantification theory of coherence. In this paper, we study this problem by use
of four important coherence measures -- the $l_1$ norm of coherence, the
relative entropy of coherence, the geometric measure of coherence and the
modified trace distance measure of coherence. We show that each pair of these
measures give a different ordering of qudit states when $d\geq 3$. However, for
single-qubit states, the $l_1$ norm of coherence and the geometric coherence
provide the same ordering. We also show that the relative entropy of coherence
and the geometric coherence give a different ordering for single-qubit states.
Then we partially answer the open question proposed in [Quantum Inf. Process.
15, 4189 (2016)] whether all the coherence measures give a different ordering