Upper Bounds for Entanglement-Assisted Classical Communication in Asymptotic Setting

Let $\mathcal{N}$ be a quantum channel. For all $\varepsilon\in[0,1)$, $n\in\mathbb{N}$, and every $(n,|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $n$ uses of $\mathcal{N}$, \[ \frac{1}{n}\log_2 |\mathcal{M}| \le \frac{1}{1-\varepsilon}\left(\frac{1}{n}I(\mathcal{N}^{\otimes n})+\frac{1}{n}h_2(\varepsilon)\right), \] and, for all $\alpha>1$, \[ \frac{1}{n}\log_2 |\mathcal{M}| \le \frac{1}{n}\widetilde{I}_{\alpha}(\mathcal{N}^{\otimes n}) + \frac{\alpha}{n(\alpha-1)}\log_2\!\left(\frac{1}{1-\varepsilon}\right). \] Using additivity, these bounds imply that $I(\mathcal{N})$ is a strong converse rate and $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})\le I(\mathcal{N})$. The single-letter strong-converse quantity is $\widetilde{I}_{\alpha}(\mathcal{N})$, and the entropy correction is $h_2(\varepsilon)$.