the phase-matching condition, therefore six home-built Fabry-Pérot cavities with high performance are employed to split and retrieve Stokes and anti-Stokes photons. As for single cavity, the transmission rate of each cavity reaches around $95\%$ , and the extinction rate of each cavity for signal and noise is up to $500:1$.

#### Witness for $M$-separability: {#witness-for-m-separability .unnumbered}

In the certifying process, we apply an optical pulse to interrogate the state of atomic ensemble by analyzing the correlated photons statistics. Firstly, we assume that the atomic state can be described in a pure state, which can be decomposed into a $M$-separable form like equation $\left |\psi  \right \rangle$ in the main text. Due to the imperfect experimental conditions and decoherence, the representation of state in every independent subgroup may be a superposition of many possible states[@Duan_25memory(2017)]. Since we consider at most two excitations in the SRS process, the specific form of each state in the subgroup can be expressed in following state $$\label{eq1}
\left |\phi _{i} \right \rangle=a_{i}\left |W _{0} \right \rangle+b_{i}\left |W _{1} \right \rangle+c_{i}\left |W _{2} \right \rangle$$ where $\left |W_{1} \right \rangle$, $\left |W_{2} \right \rangle$ are the Dicke states in each subgroup; $\left |W_{0} \right \rangle$ is a 