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Let $\mathcal{E} : \mathcal{L}(A) \to \mathcal{L}(B)$ be a completely positive trace preserving map. By the Choi–Jamiołkowski isomorphism there is an isometry $J : A \to B \otimes C$ such that $\mathcal{E}(\rho) = \textrm{Tr}_C\{J \rho J^\dagger\}$. There is also a complementary channel $\mathcal{F} : \mathcal{L}(A) \to \mathcal{L}(C)$ with action $\mathcal{F}(\rho) = \textrm{Tr}_B\{J \rho J^\dagger\}$.

I am interested in a rank four tensor $Q : \mathcal{L}(A)^{\otimes 2}$ defined as $$ \left<ij\middle|Q\middle|kl\right> = \mathrm{Tr}_B\{ \mathcal{E}(\left|k\right>\left<i\right|) \mathcal{E}(\left|l\right>\left<j\right|)\}. $$ In terms of the complementary channel, this can be written $$ \left<ij\middle|Q\middle|kl\right> = \mathrm{Tr}_C\{ \mathcal{F}(\left|k\right>\left<j\right|) \mathcal{F}(\left|l\right>\left<i\right|)\}. $$ As a trace diagram (lines represent contraction of tensor indices), $Q$ is equal to

trace diagram corresponding to above equations

I think this object should somehow encode information about the channel such as channel capacity. Has this been studied? What is known about it?

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