Canonical-purification Uhlmann theorem

For states $\rho$ and $\sigma$, let $\psi_\rho$ and $\psi_\sigma$ be the canonical purifications of $\rho$ and $\sigma$, respectively. There exists a reference unitary $U$ such that $F(\rho,\sigma)^2 = |\langle \psi_\rho | U | \psi_\sigma \rangle|^2$, and for every reference unitary $V$, $|\langle \psi_\rho | V | \psi_\sigma \rangle|^2 \le F(\rho,\sigma)^2$.