Detecting and estimating coherence based on coherence witnesses

Quantum coherence has wide-ranging applications from quantum thermodynamics to quantum metrology, quantum channel discrimination, and even quantum biology. Thus, detecting and quantifying coherence are two fundamental problems in quantum resource theory. Here, we introduce feasible methods to detect and estimate the coherence by constructing coherence witnesses for any finite-dimensional states. Our coherence witnesses detect coherent states by testing whether the expectation value of the witness is negative or not. Two typical coherence witnesses are proposed and discussed based on our witness-constructing method, which are also used to estimate the robustness of coherence, the ${l}_{1}$ norm, and the ${l}_{2}$ norm of coherence measures. Furthermore, we compare one of our coherence witnesses with a previously introduced witness by proving that our witness is strictly stronger than that previous witness. We also present an application of coherence in a quantum metrology task, in which we estimate an unknown parameter by measuring our coherence witness.