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Following your attempt we have \begin{align*} | \phi \rangle &= \sum_i \phi_i | a_i^{\phi} \rangle | b_i^{\phi} \rangle\\ | \gamma \rangle &= \sum_i \gamma_i | a_i^{\gamma} \rangle | b_i^{\gamma} \rangle, \end{align*}  \rho \equiv \operatorname{tr}_B(| \psi \rangle \langle \psi |) = |\alpha|^2 \rho_{\phi \phi} + |\beta|^2 \rho_{\gamma \...

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