Quantitative coherence witness for finite dimensional states

We define the stringent coherence witness as an observable whose mean value vanishes for all incoherent states but nonzero for some coherent states. Such witnesses are proved to exist for any finite-dimension states. Not only is the witness efficient in testing whether a state is coherent, but also its mean value can quantitatively reveal the amount of coherence. For an unknown state, the modulus of the mean value of a normalized witness provides a tight lower bound to the l1-norm of coherence. When we have some previous knowledge of a state, the optimal witness which has the maximal mean value is derived. It is proved that for any finite dimension state, the mean value of the optimal witness, which we call the witnessed coherence, equals the l1-norm of coherence. In the case that the witness is fixed and the incoherent operations are allowed, the maximal mean value can reach the witnessed coherence if and only if certain relations between the fixed witness and the initial state are satisfied. Our results provide a way to directly measure the coherence in arbitrary finite dimension states and an operational interpretation of the l1-norm of coherence.