$M$ value, we ought to determine the lower bound of the entanglement state with $D$ particles entangled (see Methods). The witness operator can be expressed as $$\label{eq5}
\omega ^{M}=\left |W_{2} \right \rangle\left \langle W_{2} \right |-p_{2}^{bound}(p_{1},M)$$ where the two key parameters $p_{1}$ and $p_{2}$ are projective probabilities in the forms of $\left | \left \langle W_{1}|\psi  \right \rangle \right |^{2}$ , $\left | \left \langle W_{2}|\psi  \right \rangle \right |^{2}$, and $p_{2}^{bound}(p_{1},M)$ stands for the theoretical minimal value of $p_{2}$ under the condition of fixed $p_{1}$ and $M$ value. For the density matrix $\rho$ of experimental state, $tr(\rho \omega ^{M} ) < 0$ means that the entanglement depth is at least $D=N/M$. In actual experiments, $p_{1}$ should be defined as the conditional probability of detecting a correlated anti-Stokes photon with a heralded forward Stokes photon, and the probability $p_{2}$ stands for the probability of two excitation events, which is deduced by the autocorrelation function $g_{AS_{1}-AS_{2}|S}^{(2)}={2p_{2}}/{p_{1}^{2}}$ measured by a Hanbury Brown-Twiss interferometer.

The decoherence effects can be revealed by the observation of entanglement depth’s evolution via adjusting the delay of optical probe pulse in our quantum memory built-in configuration. The $p_{1}$ can be influenced by the retrieval efficiency of quantum memory, 