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Let $\mathcal{N} : M_n \to M_n$ be a unital quantum channel (hence trace preserving and completely positive).

Do we have the inequality $Q^{(1)}(\mathcal{N}) \leq \chi(\mathcal{N})$ ?

Here $\chi(\mathcal{N})$ is the Holevo capacity (or Holevo information) of $T$ and $Q^{(1)}(\mathcal{N})$ is the coherent information of $\mathcal{N}$.

I ask this question since I know that $Q(\mathcal{N}) \leq C(\mathcal{N})$ where $Q(\mathcal{N})$ and $C(\mathcal{N})$ are the regularized version of $Q^{(1)}(\mathcal{N})$ and $\chi(\mathcal{N})$.

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  • $\begingroup$ Perhaps you have a typo since $Q(\mathcal{N})\geq C(\mathcal{N}) \geq \chi(\mathcal{N})$ since these are the regularized coherent information (equal to the entanglement assisted classical capacity), unassisted classical capacity and the Holevo quantity. $\endgroup$ Commented Sep 14, 2020 at 18:20

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