Estimating coherence with respect to general quantum measurements

Abstract

The conventional coherence is defined with respect to a fixed orthonormal basis, i.e., to a von Neumann measurement. Recently, generalized quantum coherence with respect to general positive operator-valued measurements has been presented. Several well-defined coherence measures, such as the relative entropy of coherence \(C_{r}\), the \(l_{1}\) norm of coherence \(C_{l_{1}}\) and the coherence \(C_{T,\alpha }\) based on Tsallis relative entropy with respect to general POVMs have been obtained. In this work, we investigate the properties of \(C_{r}\), \(C_{l_{1}}\) and \(C_{T,\alpha }\). We estimate the upper bounds of \(C_{l_{1}}\); we show that the minimal error probability of the least square measurement state discrimination is given by \(C_{T,1/2}\); we derive the uncertainty relations given by \(C_{r}\), and calculate the average values of \(C_{r}\), \(C_{T,\alpha }\) and \(C_{l_{1}}\) over random pure quantum states. All these results include the corresponding results of the conventional coherence as special cases.