# Forbidden/allowed outputs of a quantum channel

The coherent information of a channel $$\mathcal{E}_{A'\rightarrow B}$$ is defined as the maximum value obtained by the following function where the maximization is over all input states

$$I_{\rm{coh}}(\mathcal{E}) = \max_{\rho_{A'}} S(\mathcal{E}(\rho_{A'})) - S((I_A\otimes \mathcal{E})(\rho_{AA'})),$$

where $$\rho_{AA'}$$ is the purification of $$\rho_{A'}$$. I can think in terms of the complementary channel $$\mathcal{E}^c_{A'\rightarrow E}$$ instead and the optimization becomes

$$I_{\rm{coh}}(\mathcal{E}) = \max_{\rho_{A'}} S((I_A\otimes \mathcal{E^c})(\rho_{AA'})) - S(\mathcal{E^c}(\rho_{A'}))$$

Suppose I forget about the channel for a moment and only look at the registers $$A$$ and $$E$$. Clearly, the state that maximizes the coherent information is the maximally mixed state $$I_{AE}$$. In some sense, this is the ideal output state that we try to achieve. For any strictly noisy channel, it must hold that there exists no pure input $$\rho_{AA'}$$ such that

$$(I\otimes \mathcal{E^c})\rho_{AA'} = I_{AE}$$

Is there a general way to see this impossibility/possibility result for arbitrary states on the right hand side? In other words, if I am given a channel and its complementary channel (but don't know its capacity) and a "target" state $$\rho_{AE}$$, can I efficiently prove that there exists/cannot exist a pure input $$\rho_{AA'}$$ that satisfies

$$(I\otimes \mathcal{E^c})\rho_{AA'} = \rho_{AE}$$