11 f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) = I ( X ; B$_{1}$B$_{2}$ )$_{ρ}$ + λI ( A 〉 B$_{1}$B$_{2}$X )$_{ρ}$ = H ( B$_{1}$B$_{2}$ )$_{ρ}$ + ( λ − 1) H ( B$_{1}$B$_{2}$ | X )$_{ρ}$ − λH ( E$_{1}$E$_{2}$ | X )$_{ρ}$ ≤ H ( B$_{1}$ )$_{ρ}$ + ( λ − 1) H ( B$_{1}$ | X )$_{ρ}$ − λH ( E$_{1}$ | X )$_{ρ}$ + H ( B$_{2}$ )$_{ρ}$ + ( λ − 1) H ( B$_{2}$ | B$_{1}$X )$_{ρ}$ − λH ( E$_{2}$ | E$_{1}$X )$_{ρ}$ ≤ H ( B$_{1}$ )$_{ρ}$ + ( λ − 1) H ( B$_{1}$ | X )$_{ρ}$ − λH ( E$_{1}$ | X )$_{ρ}$ + H ( B$_{2}$ )$_{σ}$ + ( λ − 1) H ( B$_{2}$ | Y X )$_{σ}$ − λH ( E$_{2}$ | Y X )$_{σ}$ = [ I ( X ; B$_{1}$ )$_{θ}$ + λI ( AA$_{2}$ 〉 B$_{1}$X )$_{θ}$] + [ I ( XY ; B$_{2}$ )$_{σ}$ + λI ( AE$_{1}$ 〉 B$_{2}$XY )$_{σ}$] ≤ f$_{λ}$ ( N$_{1}$ ) + f$_{λ}$ ( N$_{2}$ ) . B. CE Trade-off Curve The first equality follows by definition and the as- sumption that ρ maximizes f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ). The sec- ond equality follows from entropic manipulations and the fact that H ( AB$_{1}$B$_{2}$ | X ) = H ( E$_{1}$E$_{2}$ | X ) for the state ρ . The first inequality follows from subadditivity of entropy and the chain rule [61]. The second inequal- ity uses two applications of the monotonicity of condi- tional entropy with respect to quantum channels acting on the conditioned system. Specifically, H ( B$_{2}$ | B$_{1}$X )$_{ρ}$ ≤ H ( B$_{2}$ | Y X )$_{σ}$ because of the existence of the map D$_{1}$ while H ( E$_{2}$ | Y X )$_{σ}$ ≤ H ( E$_{2}$ | E$_{1}$X )$_{ρ}$ because of the ex- istence of the map D$_{2}$ . The third equality follows be- cause H ( E$_{1}$ | X )$_{ρ}$ = H ( AA$_{2}$B$_{1}$ | X )$_{θ}$ and H ( E$_{2}$ | Y X )$_{ρ}$ = H ( AE$_{1}$B$_{2}$ | Y X )$_{σ}$. The final inequality follows because θ and σ are both states of the form in (15). For the CE region, we would like to maximize both the classical communication rate while minimizing the entanglement consumption rate, but we cannot have both be simultaneously optimal. Thus, we must trade between these resources. If we are willing to reduce the classical communication rate by a little, then the protocol does not require as much entanglement consumption. Our main theorem below states that the following func- tion generates points on the CE trade-off curve: where the state ρ is of the form in (15) and 0 ≤ λ ≤ 1. Theorem 3 For 0 ≤ λ ≤ 1 , the function in (23) leads to a point Corollary 1 The one-shot CQ trade-off curve is equal to the regularized CQ trade-off curve when the noisy quan- tum channel N is a Hadamard channel: g$_{λ}$ ( N ) ≡ max$_{ρ}$ I ( AX ; B )$_{ρ}$ − λH ( A | X )$_{ρ}$ , (23) ( I ( AX ; B )$_{ρ}$ ,H ( A | X )$_{ρ}$) f$_{λ}$ $^{(}$N ⊗ n ) = nf$_{λ}$ ( N ) . f$_{λ}$ ( N ⊗ n $^{+1}$) = f$_{λ}$ ( N ⊗ N ⊗ $^{n}$) = f$_{λ}$ ( N ) + f$_{λ}$ ( N ⊗ $^{n}$) = f$_{λ}$ ( N ) + nf$_{λ}$ ( N ) . on the CE trade-off curve, provided ρ maximizes (23). Proof. The proof of this theorem proceeds similarly to that of Theorem 2 in the previous section. It follows from the results of Lemmas 8 and 9 and Corollary 2 in Appendix B. Proof. We prove the result using induction on n . The base case for n = 1 is trivial. Suppose the result holds for n : f$_{λ}$ ( N ⊗ $^{n}$) = nf$_{λ}$ ( N ). The following chain of equalities then proves the inductive step: The results in the previous section demonstrate that the CQE capacity region for all Hadamard channels single-letterizes. These results imply that we can actu- ally compute the CQE capacity region for these chan- nels, by the arguments in Section III C. In this section, we consider several instances of Hadamard channels and show how we can exactly characterize their correspond- ing CQE capacity region. We first consider the qubit dephasing channel and the 1 → N cloning channel, and then show how to apply the results for the 1 → N cloning channel to the Unruh channel. There is one last point that we should address con- cerning the CQ trade-off curve. There is the possibility in this trade-off problem that the parameter λ does not parametrize all points on the trade-off curve, potentially leading to a gap in the trade-off curve. We address this concern in Appendix C by proving that one gets the en- tire trade-off curve by varying λ and taking the convex hull of the resulting points. A similar proof holds for the CE trade-off curve. The first equality follows by expanding the tensor prod- uct. The second critical equality follows from the appli- cation of Lemma 4 because N is a Hadamard channel. The final equality follows from the induction hypothesis. This section briefly recalls the parametrizations for the CQ and CE trade-off curves for a qubit dephasing chan- A. Qubit Dephasing Channel V. PARAMETRIZATION OF THE TRADE-OFF CURVES