One-Shot Lower Bounds for Entanglement-Assisted Classical Communication

Let $\mathcal{N}_{A\to B}$ be a quantum channel. For all $\varepsilon\in(0,1)$ and $\eta\in(0,\varepsilon)$, there exists an $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}_{A\to B}$ such that \[ \log_2 |\mathcal{M}| = \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N}) - \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right). \] Consequently $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})\ge \overline{I}_{H}^{\varepsilon-\eta}(\mathcal{N})-\log_2(4\varepsilon/\eta^2)$. Moreover, for all $\alpha\in(0,1)$, there exists an $(|\mathcal{M}|,\varepsilon)$ protocol satisfying \[ \log_2 |\mathcal{M}| \ge \overline{I}_{\alpha}(\mathcal{N}) - \frac{\alpha}{1-\alpha}\log_2\!\left(\frac{1}{\varepsilon-\eta}\right) - \log_2\!\left(\frac{4\varepsilon}{\eta^2}\right). \] The hypothesis-testing quantity is denoted inline by $I_H$, and the Renyi parameter appears in $\overline{I}_{\alpha}$.