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\mathcal{E}(\rho) = \mathcal{E}(\sum_ip_i\rho_i) = \sum_i \Pr\{i,\mathcal{E}\}\dfrac{\mathcal{E}(\rho_i)}{\text{tr}\{\mathcal{E}(\rho_i)\}} = \Pr\{\mathcal{E}\} \sum_i \Pr\{i|\mathcal{E}\}\dfrac{\mathcal{E}(\rho_i)}{\text{tr}\{\mathcal{E}(\rho_i)\}},