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Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{A}$. Given two isomorphic systems $\mathrm{A}$ and $\mathrm{A}'$, let $\sigma$ denote the maximally mixed state on the symmetric subspace (under permutation of $\mathrm{A}$ and $\mathrm{A}'$). I am trying to understand under which conditions on $\mathcal{N}$ the following holds: $$ [ \mathcal{N}(\pi) \otimes \mathcal{N}(\pi) , (\mathcal{N} \otimes \mathcal{N})(\sigma) ] = 0 \, . $$ For instance, this equality holds if $\mathcal{N}$ is the identical channel, or a depolarizing channel $\mathcal{N}(\rho) = p \rho + (1-p)\pi$.

Thanks a lot of your help!

PS: if you are interested, I need this in relation to this paper: http://arxiv.org/abs/1507.06038

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