vacuum state. Thus, the whole state of ensemble is the product of all subgroups: $$\label{eq2}
\left |\Psi  \right \rangle=\bigotimes_{i=1}^{M}a_{i}\left |W _{0} \right \rangle+b_{i}\left |W _{1} \right \rangle+c_{i}\left |W _{2} \right \rangle$$ The two probability $p_{1}$, $p_{2}$ can be calculated specifically, $$\label{eq3}
p_{1}=\frac{\left |\prod_{i=1}^{M}a_{i}  \right |^{2}}{M}\left | \sum_{i} \frac{b_{i}}{a_{i}}\right |^{2}$$ $$\label{eq4}
p_{2}=\frac{\left |\prod_{i=1}^{M}a_{i}  \right |^{2}}{M^{2}(1-\frac{1}{N})}\left | \sqrt{2}\sum_{i<j} \frac{b_{i}b_{j}}{a_{i}a_{j}}+\sqrt{1-\frac{1}{D}}\sum_{i} \frac{c_{i}}{a_{i}}\right |^{2}$$ where $D=\frac{N}{M}$ is the entanglement depth. Actually, the entanglement depth should be defined as the largest scale of all the entangled subgroups, however, we can take ${N}/{M}$ as the representation of entanglement depth to avoiding get vary large subgroup size[@NC_million-atoms].

In order to determine the entanglement depth in experiment, we need to determine the lower bound for $p_{2}$ with a fixed $M$ value. Obviously, we need the probability $p_{2}$ as low as possible in actual experiment, which means that the fidelity of target $W_{1}$ state is high. The bound can be calculated by $$\label{eq5}
p_{2}^{bound}(p_{1},M)=min{\left \{ p_{2}|\psi :p_{1}, M = const \right \}}$$ Note that there are constraints between the coefficients of superposition in equation (3), $\left | a_{i} \right |^{2}+\left | b_{i} \right |^{2}+\left | c_{i} \right |^{2} \leq 1 $, and we can take the approximation $|a_{i}|^{2}+|b_{i}|^{2}+|c_{i}|^{2}= 1$ owing 