Alternative definition of the coherent information of a quantum channel

Question

Let $T: M_n to M_n$ be a quantum channel. If I understand Definition 13.5.1 of the book "Quantum information theory" of Wilde, the coherent information $Q(T)=max_{phi_{AA'}} I(A rangle B)_rho$ of $T$ is given by
begin{equation}
Q(T)
=sup_{rho textrm{ pure}} Big{Hbig((mathrm{tr} otimes T)(rho)big)-Hbig((mathrm{Id} otimes T)(rho)big)Big} 
end{equation}
where the supremum is taken over all bipartite pure states $rho$ on $M_n otimes M_n$.
It is true that we can replace the supremum by a supemum on all states (not necessarily pure) ?

rnva · Answer

I follow Wilde's notation here. The coherent information of a channel $N:A' rightarrow B$ is given by
$$Q(N) equiv max_{phi_{A A^{prime}}} I(Arangle B)_{rho},$$
where $rho_{AB}=N_{A^{prime} rightarrow B}left(phi_{A A^{prime}}right)$. Notice that the channel only acts on the $A'$ register. The $A$ register is used to purify the input to the channel.
So yes, you can consider mixed states on $phi_{AA'}$ but then you can just purify this to some $phi_{RAA'}$ and relabel the $RA$ register as $A$.
Finally, a side point but note that the coherent information is obtained by taking the maximum over all input pure states, not the supremum.

Alternative definition of the coherent information of a quantum channel

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