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I am interested in bounding the effects of system-environment interaction.

Suppose I have an initial state $\rho \in \mathcal{H}_S \otimes \mathcal{H}_E$ where the system and environment might be correlated. For simplicity suppose both the system and environment are qubits. Now we subject the state to the Hamiltonian $H = H_S \otimes I + I \otimes H_E + \alpha H_{SE}$.

Similarly, consider the state $\rho_S \in \mathcal{H}_S$ evolving via the Hamiltonian $H' = H_S$. I would like a bound on $$\| \rho_S(t) -\text{Tr}_E(\rho(t)) \|$$ under some suitable norm. If this is not possible, what other ways can I quantify the effect of the environment?

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