Classifying coherence with a finite set of witnesses

Coherence is a fundamental resource in quantum information processing, which can be certified by a coherence witness. In order to detect all the coherent states, we introduce a useful concept of coherence witness and structure the set of coherence witnesses $C_{[m,M]}^d$. We present necessary and sufficient conditions of detecting quantum coherence of $d-$dimensional quantum states based on $C_{[m,M]}^d$. Moreover, we show that each coherent state can be detected by one of the coherence witnesses in a finite set. The corresponding finite set of coherence witnesses is presented explicitly, which detects all the coherent states.