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2

If the density matrices commute then there exists a joint spectral decomposition of both $\rho$ and $\sigma$. I.e. there exists a unitary $U$ and diagonal positive semidefinite matrices $D_{\rho}$ and $D_{\sigma}$ such that $$\rho = U D_{\rho} U^\dagger \qquad \text{and} \qquad \sigma = U D_{\sigma} U^\dagger.$$ Note that here you are diagonalizing two ...

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The extra transformation $U_{\mathrm{env}}$ is equivalent to choosing a new basis for the environment states $$|l\rangle\to U_{\mathrm{env}}^\dagger |l\rangle.$$ This is the same freedom as the freedom to choose the basis for $|l\rangle$; equivalently, your question is whether there is additional freedom between the choices $$\Pi_l=(I\otimes \langle l|)U(I\... 2 Given any unit vector, v\equiv |v\rangle\in\mathcal X for some finite-dimensional complex vector space \mathcal X, the operator vv^\dagger\equiv|v\rangle\!\langle v| defined by$$|v\rangle\!\langle v|\equiv v v^\dagger\in \operatorname{Lin}(\mathcal X), \\ (vv^\dagger)(w)\equiv (|v\rangle\!\langle v|)(|w\rangle) \equiv v \langle v,w\rangle, is a rank-...

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