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

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


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