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This question is related to an earlier post Zero noise extrapolation for error mitigation: Meaning of rescaled density matrix, specifically when there is no local hamiltonian evolution

If I have the following, where $\mathcal{L}$ is a superoperator $$ \frac{\partial \rho_\lambda(t)}{\partial t}=\lambda \mathcal{L}(\rho(t)) $$ Then, $$ \frac{\partial \rho_{c\lambda}(t)}{\partial t}=c\lambda \mathcal{L}(\rho(t)) $$ But, is the following true? $$ \frac{\partial \rho_{c\lambda}(t)}{\partial t}=\frac{\partial \rho_\lambda(ct)}{\partial t} $$ i.e $$ \lambda \mathcal{L}(\rho(ct))= c\lambda \mathcal{L}(\rho(t)) $$

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