  In this paper, we introduce and study unified $(r,s)$-relative entropy and
quantum unified $(r,s)$-relative entropy, in particular, our main results of
quantum unified $(r,s)$-relative entropy are established on the separable
complex Hilbert spaces. Moreover, the entanglement-measure of states due to the
quantum unified $(r,s)$-relative entropy is considered, too. Our results
improved a uncorrect statement on the monotone property of entanglement-measure
function.
