One-Shot Entanglement-Assisted Classical Capacity

Given a quantum channel $\mathcal{N}_{A\to B}$ and $\varepsilon\in[0,1]$, the one-shot $\varepsilon$-error entanglement-assisted classical capacity $C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N})$ is \[ C_{\operatorname{EA}}^{\varepsilon}(\mathcal{N}) := \sup_{(\mathcal{M},\Psi,\mathcal{E},\mathcal{D})} \{\log_2 |\mathcal{M}| : p_{\mathrm{err}}^{*}((\Psi,\mathcal{E},\mathcal{D});\mathcal{N})\le \varepsilon\}. \] The supremum ranges over one-shot entanglement-assisted classical communication protocols with message set $\mathcal{M}$, shared state $\Psi_{A'B'}$, encoding channel $\mathcal{E}_{M'A'\to A}$, and decoding channel $\mathcal{D}_{BB'\to\widehat{M}}$, with message and estimate registers indexed by the message set.