23 Proof. The proof of this lemma is similar to the proof of Lemma 9 in Ref. [71]. Consider another classical- quantum state σ in addition to the two presented in the statement of the theorem: Then the following chain of inequalities holds when 0 ≤ λ ≤ 1: in this computation: We compute the determinants of the density operators ψ E 0 and ψ E 1 : The first equality follows from entropic manipulations. The second equality follows because the system X is clas- sical and the states φ AA ′ x are pure. The inequality follows because the entropies H ( A $^{′}$) and H ( B )$_{ρ}$ can only in- crease under a complete dephasing. The third equality follows because N ◦ Δ = Δ ◦ N and N c ◦ Δ = N c for a generalized dephasing channel N , and the final equality follows from entropic manipulations. These determinants lead to the same eigenvalues for both ψ E 0 and ψ E 1 : Thus, the coherent information is as stated in the theo- rem: I ( A 〉 BX ) = H$_{2}$ ( µ ) − H$_{2}$ ( λ$_{+}$ ). We first prove two important lemmas and then prove a theorem that gives the exact form of the CE trade-off curve. where the states ψ$_{0}$ and ψ$_{1}$ are the same as they are in the statement of Lemma 13. Lemma 15 An ensemble of the following form parametrizes all points on the CE trade-off curve for a qubit dephasing channel: Proof. The proof proceeds similarly to the proof of Lemma 13, with the same definitions of states ρ and σ , but with the following different chain of inequalities for 0 ≤ λ ≤ 1: Lemma 14 Let N be a generalized dephasing channel. In the optimization task for the CE trade-off curve, it suffices to consider a classical-quantum state with diag- onal conditional density operators, in the sense that the following inequality holds when 0 ≤ λ ≤ 1 : U A $^{′}$→ BE N is an isometric extension of N , ϕ A ′ x = Δ( ϕ A ′ x ) = Δ( φ A ′ x ) , and Δ is the completely dephasing channel. We do not provide justifications for the above steps be- cause they are identical those in the proof of Lemma 13. I ( AX ; B )$_{ρ}$ − λH ( A | X )$_{ρ}$ = (1 − λ ) H ( A | X )$_{ρ}$ + H ( B )$_{ρ}$ − H ( E | X )$_{ρ}$ = (1 − λ ) H ( A | X )$_{σ}$ + H ( B )$_{ρ}$ − H ( E | X )$_{σ}$ ≤ (1 − λ ) H ( A | X )$_{σ}$ + H ( B )$_{σ}$ − H ( E | X )$_{σ}$ = (1 − λ ) H ( A | X )$_{σ}$ + 1 − H ( E | X )$_{σ}$ = 1 + ∑ x p$_{X}$ ( x ) $^{[}$(1 − λ ) H ( A )$_{ψ}$$_{x}$ − H ( E )$_{ψ}$$_{x}$ ] ≤ 1 + max$_{x}$ $^{[}$(1 − λ ) H ( A )$_{ψ}$$_{x}$ − H ( E )$_{ψ}$$_{x}$ ] = 1 + (1 − λ ) H ( A )$_{ψ}$$_{∗}$ x − H ( E )$_{ψ}$$_{∗}$ x . I ( AX ; B )$_{ρ}$ − λH ( A | X )$_{ρ}$ ≤ I ( AX ; B )$_{θ}$ − λH ( A | X )$_{θ}$ , ρ XABE ≡ ∑ x p$_{X}$ ( x ) | x 〉 〈 x | X ⊗ U A $^{′}$→ BE N ( φ AA ′ x ) , θ XABE ≡ ∑ x p$_{X}$ ( x ) | x 〉 〈 x | X ⊗ U A $^{′}$→ BE N ( ϕ AA ′ x ) , where Appendix E: Form of the CE Trade-off Curve for Qubit Dephasing Channels 1 2 ( | 0 〉 〈 0 | X ⊗ ψ AA ′ 0 + | 1 〉 〈 1 | X ⊗ ψ AA ′ 1 ) , (E1) λ$_{±}$ ≡ 1 2 ±$^{√}$ 1 4 − Det( ψ E o ) (D7) = 1 2 ± 1 2$^{√}$ 1 − 16 · p 2 $^{(}$1 − p 2$^{)}$ µ (1 − µ ) . Det( ψ E 0 ) = Det( ψ E 1 ) = $^{(}$1 − p 2$^{)}$ p 2 (1 − (2 µ − 1) $^{2}$) = 2 pµ $^{(}$1 − p 2$^{)}$ (1 − µ ) . I ( AX ; B )$_{ρ}$ − λH ( A | X )$_{ρ}$ = (1 − λ ) H ( A | X )$_{ρ}$ + H ( B )$_{ρ}$ − H ( E | X )$_{ρ}$ = (1 − λ ) ∑ x p$_{X}$ ( x ) H ( A $^{′}$)$_{φ}$$_{x}$ + H ( B )$_{ρ}$ − H ( E | X )$_{ρ}$ ≤ (1 − λ ) ∑ x p$_{X}$ ( x ) H ( A $^{′}$)$_{Δ(}$$_{φ}$$_{A}$ ′ x ) + H ( Y )$_{σ}$ − H ( E | X )$_{σ}$ = (1 − λ ) H ( A | X )$_{θ}$ + H ( B )$_{θ}$ − H ( E | X )$_{θ}$ = I ( AX ; B )$_{θ}$ − λH ( A | X )$_{θ}$ . ρ XE = 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ E 0 + | 1 〉〈 1 | X ⊗ ψ E 1 ] , ψ E 0 = $^{(}$1 − p 2$^{)}$ | 0 〉〈 0 | E + p 2 | 1 〉〈 1 | E +$^{√}$1 − p 2$^{√}$ p 2 (2 µ − 1)( | 0 〉〈 1 | E + | 1 〉〈 0 | $^{E}$) , ψ E 1 = $^{(}$1 − p 2$^{)}$ | 0 〉〈 0 | E + p 2 | 1 〉〈 1 | E −$^{√}$ 1 − p 2$^{√}$ p 2 (2 µ − 1)( | 0 〉〈 1 | E + | 1 〉〈 0 | $^{E}$) . σ XY$_{A}$Y$_{B}$E ≡ ∑ x p$_{X}$ ( x ) | x 〉 〈 x | $^{X}$⊗ Δ B → $^{Y}$( U A $^{′}$→ BE N ( φ AA ′ x )) .