Infinite operator-sum representation of density operator for a dissipative cavity with Kerr medium derived by virtue of entangled state representation

By using the thermo entangled state representation we solve the master equation for a dissipative cavity with Kerr medium to obtain density operators' infinite operator-sum representation}$\rho (t) =\sum_{m,n,l=0}^{\infty}M_{m,n,l}\rho_{0}\mathcal{M}_{m,n,l}^{\dagger}.$ It is noticeable that}$M_{m,n,l}$ is not hermite conjugate to $\mathcal{M}_{m,n,l}^{\dagger}$, nevertheless the normalization}$\sum_{m,n,l=0}^{\infty}\mathcal{M}_{nm,,l}^{\dagger}M_{m,n,l}=1$ still holds}, i.e., they are trace-preserving in a general sense. This example may stimulate further studying if general superoperator theory needs modification.