10 classical capacity because I ( XY ; B )$_{σ}$ ≤ max$_{ρ}$ I ( X ; B ). It then follows that the trade-off curve really starts when λ ≥ 1. Proof. This theorem follows from the results of Lem- mas 2, 3, and 4 and Corollary 1 below. Lemma 2 For any fixed λ ≥ 0 , the function in (21) leads to a point ( I ( X ; B )$_{ρ}$ , I ( A 〉 BX )$_{ρ}$) on the one-shot CQ trade-off curve in the sense of Theorem 2. We remark that the above proof gives an alternate mathemathical justification for considering only λ ≥ 1 in the maximization than does the original operational reason given in Ref. [31]. The following lemma is the crucial one that leads to our main result in this section: the single-letterization of the CQ trade-off curve for Hadamard channels. Proof. We argue by contradiction. Suppose that a particular state ρ XAB of the form in (15) maximizes (21). Suppose further that it does not lead to a point on the trade-off curve. That is, given the constraint Q = I ( A 〉 BX )$_{ρ}$, there is some other state σ XAB of the form in (15) such that I ( A 〉 BX )$_{σ}$ = I ( A 〉 BX )$_{ρ}$ = Q where C = I ( X ; B )$_{σ}$ > I ( X ; B )$_{ρ}$. But this result implies the following inequality for all λ ≥ 0: Lemma 4 The following additivity relation holds for a Hadamard channel N$_{1}$ and any other channel N$_{2}$ : contradicting the fact that the state ρ XAB maximizes (21). Proof. The inequality f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) ≥ f$_{λ}$ ( N$_{1}$ ) + f$_{λ}$ ( 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 f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) ≤ f$_{λ}$ ( N$_{1}$ ) + f$_{λ}$ ( N$_{2}$ ) when N$_{1}$ is a Hadamard channel. Lemma 3 We obtain all points on the one-shot CQ trade-off curve by considering λ ≥ 1 in the maximiza- tion in (21) because the maximization optimizes only the classical capacity for all λ such that 0 ≤ λ < 1 . Proof. Consider a state ρ XABE of the form in (15). Suppose that we perform a von Neumann measurement of the system A , resulting in a classical variable Y , and let σ XY BE denote the resulting state. Then the following chain of inequalities holds for all λ such that 0 ≤ λ < 1: f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ) = f$_{λ}$ ( N$_{1}$ ) + f$_{λ}$ ( N$_{2}$ ) . I ( X ; B )$_{σ}$ + λI ( A 〉 BX )$_{σ}$ > I ( X ; B )$_{ρ}$ + λI ( A 〉 BX )$_{ρ}$ , Suppose that N A$_{1}$ → B$_{1}$ 1 is a Hadamard channel and N A$_{2}$ → B$_{2}$ 2 is any quantum channel with respective com- plementary channels ( N c $_{1}$) A$_{1}$ → E$_{1}$ and ( N c $_{2}$) A$_{2}$ → $^{E$_{2}$}$. Then N c 1 is entanglement-breaking because N$_{1}$ is a Hadamard channel. Therefore, there exists a degrading map D B$_{1}$ → E$_{1}$ because N c 1 is entanglement-breaking. The de- grading map consists of a measurement with classical output on system Y followed by the preparation of a state on E$_{1}$ depending on the measurement outcome (as described in Section II C 2). We can therefore write D B$_{1}$ → E$_{1}$ = D Y → E$_{1}$ 2 ◦ D B$_{1}$ → Y 1 . Let where U$_{j}$ A$_{j}$ → B$_{j}$E$_{j}$ is the isometric extension of N$_{j}$ . Sup- pose further that ρ is the state that maximizes f$_{λ}$ ( N$_{1}$ ⊗ N$_{2}$ ). Also, let σ ≡ D$_{1}$ ( ρ ) and note that this state is of the following form: The first equality follows from definitions and because the state ρ XABE on systems A , B , and E is pure when con- ditioned on the classical variable X . The second equality follows because the von Neumann measurement does not affect the systems involved in the entropic expressions. The inequality follows because 0 ≤ λ < 1 and condition- ing reduces entropy. The third equality follows because the reduced state of σ XY BE on systems B and E is pure when conditioned on both X and Y . The last two equali- ties follow from algebra and the definition of the quantum mutual information. Thus, it becomes clear that the maximization of the original quantity for 0 ≤ λ < 1 is always less than the Then the following inequalities hold for all λ ≥ 1: ψ XAA$_{1}$A$_{2}$ ≡ ∑ x p$_{X}$ ( x ) | x 〉〈 x | X ⊗ | φ$_{x}$ 〉〈 φ$_{x}$ | AA$_{1}$$^{A$_{2}$}$, θ XAB$_{1}$E$_{1}$A$_{2}$ ≡ U$_{1}$ψU † $_{1}$, ρ XAB$_{1}$E$_{1}$B$_{2}$E$_{2}$ ≡ ( U$_{1}$ ⊗ U$_{2}$ ) ψ ( U$_{1}$ ⊗ U$_{2}$ ) $^{†}$, (22) σ XY AB$_{2}$E$_{1}$E$_{2}$ ≡ ∑ x p$_{X}$ ( x ) p$_{Y}$$_{|}$$_{X}$ ( y | x ) | x 〉〈 x | $^{X}$⊗ | y 〉〈 y | Y ⊗| φ$_{x,y}$ 〉〈 φ$_{x,y}$ | AB$_{2}$E$_{1}$$^{E$_{2}$}$. I ( X ; B )$_{ρ}$ + λI ( A 〉 BX )$_{ρ}$ = H ( B )$_{ρ}$ + ( λ − 1) H ( B | X )$_{ρ}$ − λH ( E | X )$_{ρ}$ = H ( B )$_{σ}$ + ( λ − 1) H ( B | X )$_{σ}$ − λH ( E | X )$_{σ}$ ≤ H ( B )$_{σ}$ + ( λ − 1) H ( B | XY )$_{σ}$ − λH ( E | XY )$_{σ}$ . = H ( B )$_{σ}$ + ( λ − 1) H ( B | XY )$_{σ}$ − λH ( B | XY )$_{σ}$ = H ( B )$_{σ}$ − H ( B | XY )$_{σ}$ = I ( XY ; B )$_{σ}$ .