We show the properties and characterization of coherence witnesses. We show methods for constructing coherence witnesses for an arbitrary coherent state. We investigate the problem of finding common coherence witnesses for certain class of states. We show that finitely many different witnesses W1,W2,⋯,Wn can detect some common coherent states if and only if ∑i=1ntiWi is still a witnesses for any nonnegative numbers ti(i=1,2,⋯,n). We show coherent states play the role of high-level witnesses. Thus, the common state problem is changed into the question of when different high-level witnesses (coherent states) can detect the same coherence witnesses. Moreover, we show a coherent state and its robust state have no common coherence witness and give a general way to construct optimal coherence witnesses for any comparable states.
Keywords: common coherence witnesses; common coherent states; high-level witnesses.