to the neglectable higher excitations. Utilizing the Lagrange multiplier method deduced in the supplementary notes of [@NC_million-atoms], the conclusion is that the symmetric solution gives the global minimal value of $ p_{2}$ for $M\leq5$. This symmetric solution requires that $a_{i}=a, b_{i}=b, c=-\sqrt{1-a^{2}-b^{2}}$. In this condition, the optimal values for $p_{1}, p_{2}$ are the solution as follows $$\label{eq6}
p_{1}^{sym}=Ma^{2M-2}b^{2}$$ $$\label{eq7}
p_{2}^{sym}=a^{2M}(\frac{1}{\sqrt{2}}(M-1)\frac{b^{2}}{a^{2}}+\frac{c}{a})^{2}$$ The final form of function $p_{2}^{bound}(p_{1},M)$ is, $$\label{eq8}
p_{2}^{bound}(p_{1},M)=a^{2M}[\frac{1}{\sqrt{2}}\frac{p_{1}(M-1)}{M}a^{-2M}-\frac{1}{a}\sqrt{1-a^{2}-\frac{p_{1}}{M}a^{2-2M}}]^{2}$$ For fixed $p_{1}$ and $ M$, we need to calculate minimal value of $p_{2}$ with $a\in (0,1)$. The theoretical bound of equation (10) can be obtained by take all values of $p_{1}\in (0,1)$, which is shown in Figure 2a and Figure 3b with different $M$ values.

#### Number of atoms involved in the creation of multipartite entanglement: {#number-of-atoms-involved-in-the-creation-of-multipartite-entanglement .unnumbered}

Theoretically, the transmission rate of probe light passing through the atomic ensemble has the following form [@phdOxford_Sprague; @absorption_model] $$\label{eq6}
T(\omega )=exp\left \{-\frac{2\pi nkLd^{2}}{h\varepsilon _{0}}\sum_{i=1}^{3}S_{i}l_{i}(\omega ) \right \}$$ where $n$ is the density of atoms, $k$ is the wave vector of probe light, $L$ is the length of our vapor cell, $d$ is the reduced dipole matrix element, $S_{i}$ is the strength of relative coupling from the hyperfine level $F=3$ of the ground 