There is a well-known theorem, firstly obtained by Denes Petz, in quantum information theory, which is described as follows:
$\mathbf{Theorem.}$ Let $\rho$ and $\sigma$ be two states on $\mathcal H$, $\Phi$ be a quantum channel defined over $\mathcal H$. If $\mathrm{supp}(\rho)\subseteq\mathrm{supp}(\sigma)$, then \begin{eqnarray} S(\rho||\sigma) = S(\Phi(\rho)||\Phi(\sigma))\quad\text{if and only if}\quad \Phi^\dagger_\sigma\circ\Phi(\rho) = \rho, \end{eqnarray} where \begin{eqnarray} \Phi^\dagger_\sigma(*) = \sigma^{1/2}\Phi^\dagger(\Phi(\sigma)^{-1/2} * \Phi(\sigma)^{-1/2}) ) \sigma^{1/2}. \end{eqnarray}
This theorem deals with the saturation of the monotonicity inequality of relative entropy.
My question is: Given arbitrary two states $\rho,\sigma$ on $\mathcal H$, can we estimate the fidelity $F(\rho,\Phi^\dagger_\sigma\circ\Phi(\rho))$ between two states $\rho$ and $\Phi^\dagger_\sigma\circ\Phi(\rho)$? Here for $\Phi^\dagger_\sigma$, you are referred to the above definition.
The notations mentioned above are explained as follows:
(1) $\mathcal H$ is a finite-dimensional Hilbert space.
(2) a state means a positive semi-definite operator on $\mathcal H$ with a unite trace such as $\rho,\sigma$.
(3) $\Phi$ is a trace-preserving completely positive map, i.e. quantum channel.
(4) $\mathrm{Ad}_M$ is a conjugate map, defined by $\mathrm{Ad}_M(X) = MXM^\dagger$.
(5) the notation $\mathrm{supp}(\rho)$ is the support space of the state $\rho$.
(6) The relative entropy between two states $\rho$ and $\sigma$ on $\mathcal H$ is defined as: $$ S(\rho||\sigma) = \mathrm{Tr}(\rho(\log\rho - \log\sigma)), $$ where $\mathrm{supp}(\rho)\subseteq \mathrm{\sigma}(\sigma)$.
(7) The fidelity between two states $\rho$ and $\sigma$ is defined by $$ F(\rho,\sigma) = \mathrm{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}). $$
(8) The operator power is taken on the support of this operator.
Another questions:
In fact, we can also estimate the relative entropies of both states: $$ S(\rho||\Phi^\dagger_\sigma\circ\Phi(\rho))\qquad S(\Phi^\dagger_\sigma\circ\Phi(\rho)||\rho). $$ Or the trace-distance $\|\rho - \Phi^\dagger_\sigma\circ\Phi(\rho))\|_1$.