Computable lower bound of the parameterized entanglement monotone

Although numerous measures of entanglement have been proposed so far, the calculation of a given faithful entanglement measure is a hard work since it is always involved in some optimization process. It is, therefore, important to estimate the lower bound of a given entanglement measure for an arbitrary quantum state. This results in a subject of intensive mathematical research. In particular, along this line, the lower bounds of concurrence or other measures that are induced from concurrence have been explored a lot. Here, we investigate the lower bounds of two kinds of entanglement monotones, i.e., $q$-concurrence ($q>1$) and $\alpha$-concurrence ($0<\alpha<1$), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete ($N$, $M$)-positive operator-valued measure [($N$, $M$)-POVM], the lower bounds for the case of $\frac12<\alpha<1$, $1<q<2$ for two-qudit states, and the case of $2\leqslant q<3$ for two-qubit states. We list several examples which show that the lower bounds based on ($N$, $M$)-POVM outperform that of GSIC-POVM and SIC-POVM, and all these measurement based bounds are better then the ones induced by positive partial transpose (PPT) and realignment criteria in literature. In addition, we obtain an analytical formula of the parameterized entanglement monotone with $\frac12<\alpha<1$ and $1<q<2$ for the isotropic state.