20 of inequalities then holds for all λ > 1: I ( AX ; B )$_{ρ}$ − λH ( A | X )$_{ρ}$ = H ( A | X )$_{ρ}$ − H ( E | X )$_{ρ}$ + H ( B )$_{ρ}$ − λH ( A | X )$_{ρ}$ = (1 − λ ) H ( BE | X )$_{ρ}$ − H ( E | X )$_{ρ}$ + H ( B )$_{ρ}$ = (1 − λ ) H ( BE | X )$_{σ}$ − H ( E | X )$_{σ}$ + H ( B )$_{σ}$ ≤ H ( B )$_{σ}$ − H ( E | X )$_{σ}$ ≤ H ( B )$_{σ}$ − H ( E | XY )$_{σ}$ = H ( B )$_{σ}$ − H ( B | XY )$_{σ}$ = I ( XY ; B )$_{σ}$ . The first equality follows from definitions and from the equality H ( AB | X )$_{ρ}$ = H ( E | X )$_{ρ}$ for the state ρ $^{XABE}$, and the second equality follows from algebra and the fact that H ( BE | X )$_{ρ}$ = H ( A | X )$_{ρ}$. The third equality follows because the von Neumann measurement does not affect the systems in the entropic quantities. The first inequal- ity follows because λ > 1 and the entropy H ( BE | X )$_{ρ}$ is always positive. The second inequality follows because conditioning reduces entropy. The fourth equality fol- lows because the reduced state of σ XY BE on systems B and E is pure when conditioned on both X and Y . The last equality follows from the definition of the quantum mutual information. Thus, it becomes clear that the maximization of the original quantity when λ > 1 is always less than the classical capacity because I ( XY ; B )$_{σ}$ ≤ max$_{ρ}$ I ( X ; B ). It then follows that the trade-off curve really occurs for the interval 0 ≤ λ ≤ 1. The following lemma is the crucial one that leads to our main result in this section: the single-letterization of the CE trade-off curve for Hadamard channels. Lemma 10 The following additivity relation holds for a Hadamard channel N$_{1}$ and any other channel N$_{2}$ : g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) = g$_{λ}$ ( N$_{1}$ ) + g$_{λ}$ ( N$_{2}$ ) Proof. The inequality g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) ≥ g$_{λ}$ ( N$_{1}$ ) + g$_{λ}$ ( N$_{2}$ ) trivially holds for all quantum channels, because the maximization on the RHS is a restriction of the max- imization in the LHS to a tensor product of states of the form in (15). Therefore, we prove the non-trivial in- equality g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) ≤ g$_{λ}$ ( N$_{1}$ ) + g$_{λ}$ ( N$_{2}$ ) when N$_{1}$ is a Hadamard channel. The channels N A$_{1}$ → B$_{1}$ 1 and N A$_{2}$ → B$_{2}$ 2 and their respective complementary channels ( N c $_{1}$) A$_{1}$ → E$_{1}$ and ( N c $_{2}$) A$_{2}$ → E$_{2}$ have the same properties as they do in the proof of Theorem 1. The state that is the output of the channels is the state ρ XAB$_{1}$E$_{1}$B$_{2}$E$_{2}$ in (22), but we now define it to be the state that maximizes g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ). Define θ as before and σ again to be the state after pro- cessing system B$_{1}$ of ρ with D$_{1}$ . The following chain of inequalities holds when λ ≤ 1: g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) = I ( AX ; B$_{1}$B$_{2}$ )$_{ρ}$ − λH ( A | X )$_{ρ}$ = (1 − λ ) H ( B$_{1}$B$_{2}$E$_{1}$E$_{2}$ | X )$_{ρ}$ − H ( E$_{1}$E$_{2}$ | X )$_{ρ}$ + H ( B$_{1}$B$_{2}$ )$_{ρ}$ = (1 − λ ) H ( B$_{1}$E$_{1}$ | X )$_{ρ}$ − H ( E$_{1}$ | X )$_{ρ}$ + H ( B$_{1}$ )$_{ρ}$ + (1 − λ ) H ( B$_{2}$E$_{2}$ | B$_{1}$E$_{1}$X )$_{ρ}$ − H ( E$_{2}$ | E$_{1}$X )$_{ρ}$ + H ( B$_{2}$ | B$_{1}$ )$_{ρ}$ ≤ (1 − λ ) H ( B$_{1}$E$_{1}$ | X )$_{ρ}$ − H ( E$_{1}$ | X )$_{ρ}$ + H ( B$_{1}$ )$_{ρ}$ + (1 − λ ) H ( B$_{2}$E$_{2}$ | Y X )$_{σ}$ − H ( E$_{2}$ | Y X )$_{σ}$ + H ( B$_{2}$ )$_{σ}$ = (1 − λ ) H ( AA$_{2}$ | X )$_{θ}$ − H ( AA$_{2}$B$_{1}$ | X )$_{θ}$ + H ( B$_{1}$ )$_{θ}$ + (1 − λ ) H ( AE$_{1}$ | Y X )$_{σ}$ − H ( AE$_{1}$B$_{2}$ | Y X )$_{σ}$ + H ( B$_{2}$ )$_{σ}$ = [ I ( AA$_{2}$X ; B$_{1}$ )$_{θ}$ − λH ( AA$_{2}$ | X )$_{θ}$] + [ I ( AE$_{1}$XY ; B$_{2}$ )$_{σ}$ − λH ( AE$_{1}$ | XY )$_{σ}$] ≤ g$_{λ}$ ( N$_{1}$ ) + g$_{λ}$ ( N$_{2}$ ) . The first equality follows because ρ is the state that maximizes g$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ). The second equality follows from entropic manipulations. The third equality fol- lows from the chain rule. The first and crucial inequal- ity follows from monotonicity of the conditional entropy H ( B$_{2}$E$_{2}$ | B$_{1}$E$_{1}$X )$_{ρ}$ under the map D$_{1}$ and the discard- ing of system E$_{1}$ , monotonicity of the conditional entropy H ( E$_{2}$ | E$_{1}$X )$_{ρ}$ under the map D$_{2}$ , and conditioning does not increase entropy. The fourth equality follows because the reduced state of θ on systems A , A$_{2}$ , B$_{1}$ , and E$_{1}$ is pure when conditioned on X , and the reduced state of σ on systems A , E$_{1}$ , B$_{2}$ , and E$_{2}$ is pure when condi- tioned on both X and Y . The fifth equality follows from Proof. The proof exploits the same induction tech- nique that Corollary 1 does, but it applies the result of Lemma 10. g$_{λ}$ $^{(}$N ⊗ n ) = ng$_{λ}$ ( N ) . Corollary 2 The one-shot CE trade-off curve is equal to the regularized CE trade-off curve when the noisy quan- tum channel N is a Hadamard channel: entropic manipulations, and the final inequality follows because θ and σ are both states of the form in (15).