Let a quantum channel $\Phi(\cdot)$ between two Hilbert spaces $\mathcal{H}_{in}$ and $\mathcal{H}_{out}$. What is the quantum channel $\Phi_{inv}(\cdot)$ that best reverses $\Phi(\cdot)$ ?
$\forall $ states $\rho$, the state $\widetilde{\rho} \equiv \Phi_{inv}(\Phi(\rho))$ should be the best approximation of $\rho$ (according to some appropriate distance measure).
If we assume that the input state comes from a known probability distribution, according to the quantum data processing inequality there are cases, depending on the source entropy and the channel properties that allow for perfect decoding.
However, how do you perform decoding in cases where perfect decoding is not possible, and in particular if no source distribution is known (so you can't assume that the state has been encoded with an error-correcting code)? I presume somebody has already thought about this, but I can't find it in the literature.