Results on various types of quantum fidelities
Theorem 1. For the Uhlmann-Jozsa fidelity the relation
\[
\begin{align}
F(\varrho,\sigma)=\left(\max_{\{p_k,\Psi_k,\Phi_k\}}\sum_k p_k |\bra{\Psi_k} \Phi_k\rangle |\right)^2\label{eq:Fidelity}
\end{align}
\]
holds such that the decompositions of \(\varrho\) and \(\sigma\) are
\[
\begin{align}
\varrho=\sum_k p_k \ketbra{\Psi_k},\quad\quad\quad \sigma=\sum_k p_k \ketbra{\Phi_k}.\label{eq:marginalcond_a}
\end{align}
\]
These states can be seen as the marginals of the bipartite separable state
\[
\begin{align}
\varrho_{12}=\sum_k p_k \ketbra{\Psi_k} \otimes \ketbra{\Phi_k}.\label{eq:sep}
\end{align}
\]
Note that the optimization is over decompositions, and not over purifications.
Note also that, without loss of generality, \(p_k\) is the same in both definitions in Eq. \eqref{eq:marginalcond_a}.
After discussing the Uhlmann-Jozsa fidelity, let us now consider the SWAP-fidelity given in [Friedland et al., Phys. Rev. Lett. 129, 110402] as
\[
\begin{align}\label{eq:Fs}
F_{S}(\varrho,\sigma)=\max_{\varrho_{12}\in \mathrm{\mathcal{D}}} {\rm Tr}(\varrho_{12} S),
\end{align}
\]
where \(S\) is the SWAP operator, and \(\mathrm{\mathcal{D}}\) is the set of quantum states.
Let us now consider a modified definition based on an optimization over separable states as
\[
\begin{align}\label{eq:Fssep_S}
F_{S,{\rm sep}}(\varrho,\sigma)=\max_{\varrho_{12}\in \mathrm{Sep}} {\rm Tr}(\varrho_{12} S),
\end{align}
\]
where \(\mathrm{Sep}\) is the set of separable states. Clearly
\[
\begin{align}
F_{S,{\rm sep}}(\varrho,\sigma)\le F_{S}(\varrho,\sigma).
\end{align}
\]
Theorem 3. For the maximum over separable states
\[
\begin{align}
F_{S,{\rm sep}}(\varrho,\sigma)=\max_{\{p_k,\Phi_k,\Psi_k\}}\sum_k p_k | \bra{\Phi_k} \Psi_k\rangle |^2\label{eq:Fssep}
\end{align}
\]
holds, where the conditions for the marginals given in Eq. \eqref{eq:marginalcond_a} are fulfilled.
From Eqs. \eqref{eq:Fidelity} and \eqref{eq:Fssep}, it is also clear that
\[
\begin{align}
F(\varrho,\sigma)\le F_{S,{\rm sep}}(\varrho,\sigma).\label{eq:FFFlele}
\end{align}
\]
Theorem 4. The SWAP-fidelity when the optimization is taken over separable states is bounded from above by the superfidelity
\[
\begin{split}
&F_{S,{\rm sep}}(\varrho,\sigma)\le F_{\rm super}(\varrho,\sigma),\label{eq:Fsuperbound}
\end{split}
\]
where
\[
\begin{align}
F_{\rm super}(\varrho,\sigma)={\rm Tr}(\varrho\sigma)+\sqrt{[1-{\rm Tr}(\varrho^2)][1-{\rm Tr}(\sigma^2)]}.\label{eq:Fsuper}
\end{align}
\]
As a consequence, for \(d=2\) (i.e., for qubits), we have
\[
\begin{align}
F_{S,{\rm sep}}(\varrho,\sigma)=F(\varrho,\sigma).\label{eq:FseqF}
\end{align}
\]
Thus, we have an explicit formula for computing the expression given in Eq. \eqref{eq:Fssep} for qubits.
[1] G. Tóth and J. Pitrik, Quantum Wasserstein distance and its relation to several types of fidelities, arXiv:2506.14523.