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Given two states $\rho_A, \sigma_A$, Uhlmann's theorem states that the fidelity between them is achieved in the following way

$$F(\rho_A, \sigma_A) = \max_{U_{R'}}F(\rho_{AR'}, (I\otimes U_{R'})\sigma_{AR'})$$

where $U_{R'}$ are unitary operators and $\rho_{AR'}, \sigma_{AR'}$ are arbitrary purifications.

Instead of purifications, let us consider extensions of the state. That is, let $\rho_{AR}$ (possibly a mixed state) satisfy $\text{Tr}_R(\rho_{AR}) = \rho_A$ and $\sigma_{AR}$ (possibly a mixed state) satisfy $\text{Tr}_R(\sigma_{AR}) = \sigma_A$. I am aware of the following fact (5.33 of these notes): For any fixed extension $\rho_{AR}$ we have $$F(\rho_A, \sigma_A) = \max_{\sigma_{AR}}\{F(\rho_{AR}, \sigma_{AR}) : \text{Tr}_R(\sigma_{AR}) = \sigma_A\} \tag{1}$$ Let $\sigma^*_{AR}$ be the state that achieves the maximum above.

Questions

  1. How is $\sigma^*_{AR}$ related to an arbitrary extension $\sigma_{AR}$? If they were purifications, one had the simple fact that they were related by a unitary on $R$.

  2. If $|R|$ was large enough that there exist pure extension states $\sigma_{AR}$, then is $\sigma^*_{AR}$, the extension that achieves the maximum in $(1)$, also pure?

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Let's start with the second question. There is nothing special about an extension $\sigma_{AR}^{\ast}$ that allows it to be optimal for the right-hand side of (1); any extension $\sigma_{AR}$ of $\sigma_A$ could happen to be optimal for the right choice of $\rho_{AR}$. For example, if we suppose that $\sigma_{AR}$ is any given extension of $\sigma_A$, and we take $\rho_{AR} = \sigma_{AR}$, then the unique optimizing extension is $\sigma_{AR}^{\ast} = \sigma_{AR}$. We may therefore not conclude that this optimal extension is pure.

For the first question, now that we know that an arbitrary choice of an extension $\sigma_{AR}$ could maximize the right-hand side of (1) for an appropriate choice of $\rho_{AR}$, we see that the question is equivalent to asking how any two extensions of the same state relate. I suppose you could come up with alternative ways to characterize the condition that two states are extensions of the same state, but there's nothing beyond this to say, and in particular there is no unitary or isometric equivalence in general when the states are not pure.

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