One-Shot Upper Bounds for Entanglement-Assisted Classical Communication

Let $\mathcal{N}_{A\to B}$ be a quantum channel and let $\varepsilon\in[0,1)$. For every $(|\mathcal{M}|,\varepsilon)$ entanglement-assisted classical communication protocol over $\mathcal{N}$, \[ \log_2 |\mathcal{M}| \le \frac{1}{1-\varepsilon}\bigl(I(\mathcal{N})+h_2(\varepsilon)\bigr), \] and, for all $\alpha>1$, \[ \log_2 |\mathcal{M}| \le \widetilde{I}_{\alpha}(\mathcal{N}) + \frac{\alpha}{\alpha-1}\log_2\!\left(\frac{1}{1-\varepsilon}\right). \] Consequently the same two upper bounds hold for $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$. The two information quantities are $I(\mathcal{N})$ and $\widetilde{I}_{\alpha}(\mathcal{N})$, and the binary entropy term is $h_2(\varepsilon)$.