Entanglement-Assisted Classical Capacity

For every quantum channel $\mathcal{N}$, its entanglement-assisted classical capacity $C_{\operatorname{EA}}(\mathcal{N})$ and its strong converse entanglement-assisted classical capacity $\widetilde{C}_{\operatorname{EA}}(\mathcal{N})$ are both equal to the mutual information $I(\mathcal{N})$: \[ C_{\operatorname{EA}}(\mathcal{N}) = \widetilde{C}_{\operatorname{EA}}(\mathcal{N}) = I(\mathcal{N}). \] Here $I(\mathcal{N}) := \sup_{\psi_{RA}} I(R;B)_{\omega}$ with $\omega_{RB}=\mathcal{N}_{A\to B}(\psi_{RA})$. This is the final capacity identity for the information quantity $I(\mathcal{N})$.