Fig.1

(Color online) The $y$ axis is the lower bound of the concurrence $\mathcal{C}^\beta_{A|BC}$. The red (green) line represents the lower bound from our result for $k=0.6$ ($k=0.8$), and the blue line represents the lower bound of Eq.(5) from Ref.[22]."

Fig.2

(Color online) The $y$ axis is the lower bound of the EoF $E^\beta_{A|BC}$. The red (green resp. blue) line represents the lower bound from our result for $k=0.5$ ($k=0.7$ resp. $k=0.9$), and the yellow line represents the lower bound from the result in Ref.[22]."

Fig.3

(Color online) The $y$ axis is the lower bound of the negativity $\mathcal{N}_c(|\psi\rangle_{A|BC})$, which are functions of $\beta$."

Fig.4

(Color online) The $y$ axis is the lower bound of the Tsallis-$q$ entanglement $T_q^\beta(|\psi\rangle_{A|BC})$. The red (green resp. yellow) line represents the lower bound from our result for $k=0.5$ ($k=0.7$ resp. $k=0.9$), and the blue line represents the lower bound from the result in Ref.[22]."

Fig.5

(Color online) The $y$ axis is the lower bound of the Rényi entropy entanglement $E^\beta_{2}(|\psi\rangle_{A|BC})$. The purple line represents the value of $E^\beta_2(|\psi\rangle_{A|BC})$, the red (green resp. yellow) line represents the lower bound from our result for $k=0.5$ ($k=0.7$ resp. $k=0.9$), and the blue line represents the lower bound from the result (55) in Ref.[14]."