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Consider a density matrix $\rho(U)$ which depends on $U \in SU(2^n)$, corresponding to a state of a composite, finite-dimensional Hilbert space $\mathcal{H} \cong \bigotimes_{i=1}^{2^n} \mathbb{C}^2$ where the first factor is designated system $\mathcal{H}_S$ and the rest of the factors is designated environment $\mathcal{H}_\mathcal{E}$. Let $dU$ denote the Haar measure (together with the infinitesimal).

Does $$ \int dU \ \text{Tr}_\mathcal{E}(\rho(U)) = \text{Tr}_\mathcal{E}(\int dU \ \rho(U))? $$

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Yes. We can check this explicitly as follows. $$\int dU \ \text{Tr}_\mathcal{E}(\rho(U)) = \int dU \ \sum_i (\mathbb{I} \otimes \langle i \lvert) \rho(U) (\lvert i \rangle \otimes \mathbb{I})\\ = \sum_i \int dU \ (\mathbb{I} \otimes \langle i \lvert) \rho(U) (\lvert i \rangle \otimes \mathbb{I})\\ = \sum_i (\mathbb{I} \otimes \langle i \lvert) \int dU \ \left( \rho(U) (\lvert i \rangle \otimes \mathbb{I}) \right)\\ = \sum_i (\mathbb{I} \otimes \langle i \lvert) \left(\int dU \ \rho(U) \right)(\lvert i \rangle \otimes \mathbb{I})\\ = \text{Tr}_\mathcal{E}(\int dU \ \rho(U)). \ \square$$

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    $\begingroup$ I think you can be more general and say all integrations commute with all traces $\endgroup$ Jul 24, 2023 at 15:02

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