Estimating coherence with respect to general quantum measurements

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-l1 and the coherence C-T,C-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-l1 and C-T,C-alpha. We estimate the upper bounds of C-l1; we show that the minimal error probability of the least square measurement state discrimination is given by C-T,C-1/2; we derive the uncertainty relations given by C-r, and calculate the average values of C-r, C-T,C-alpha and C-l1 over random pure quantum states. All these results include the corresponding results of the conventional coherence as special cases.