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Suppose I have the following Lindblad master equation: $$ \frac{d\rho}{dt}=\mathcal{L}[\rho]=-\frac{1}{2}\left(L^\dagger L \rho+\rho L^\dagger L-2L\rho L^\dagger\right) $$ where $\rho$ is a single-spin density matrix (i.e., a 2$\times$2 matrix), and $L$ is the 2$\times$2 jump operator, and $\mathcal{L}$ is a 4$\times$4 superoperator. Now, if I have a two-spin system represented by $\rho'$ (i.e., now $\rho'$ is a 4$\times$4 density matrix) such that $$ \frac{d\rho'}{dt}=\mathcal{L}'[\rho']=-\frac{1}{2}\left(L'^\dagger L' \rho'+\rho' L'^\dagger L'-2L'\rho' L'^\dagger\right) $$ where the jump operator still acts on the first spin (i.e., I have $L'=L\otimes \mathbb{I}$). How would then the superoperator $\mathcal{L}'$ is related to $\mathcal{L}$?

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The superator $\mathcal L'$ would be defined on untangled states $\rho' = \rho_1 \otimes \rho_2$ by : $$\mathcal L'[\rho_1 \otimes \rho_2] = \mathcal L[\rho_1]\otimes \rho_2$$ and extend by linearity to a general density matrix $\rho'$.

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