24 where λ N i ( µ ) ≡ ( N − 2 i ) µ + i . We then compute the entropies H ( B ) and H ( B | X ): Proof of CE trade-off in Theorem 4. The proof follows by noting that I ( AX ; B ) = H ( A | X ) + H ( B ) − H ( E | X ), H ( A | X ) = H$_{2}$ ( µ ), and that we have already computed H ( B ) and H ( E | X ) in the first part of the proof of Theorem 4. Proof. The input state in (24) that traces out the CQ curve for the cloning channels also does so for the Unruh channel. We consider the purification of this state, so that the output state is as follows: Observe that the following relationships hold: Let N ( | 0 〉 〈 0 | ) and N ( | 1 〉 〈 1 | ) denote the Unruh chan- nel outputs corresponding to the respective input states | 0 〉〈 0 | A ′ and | 1 〉〈 1 | A $^{′}$: These relationships allow us to rewrite the expression in (F1) for H ( B | X ) as follows: H ( B )$_{ρ}$ = − ∞ ∑ l =2 l − 1 ∑ i =0 p$_{l}$ ( z ) l log$^{(}$ p$_{l}$ ( z ) l ) = − ∞ ∑ l =2 p$_{l}$ ( z ) log$^{(}$ p$_{l}$ ( z ) l ) , H ( B | X )$_{ρ}$ = 1 2 $^{(}$H ( B )$_{ψ}$$_{0}$ + H ( B )$_{ψ}$$_{1}$ ) = − ∞ ∑ l =2 l − 1 ∑ i =0 p$_{l}$ ( z ) λ l − 1 i ( µ ) Δ$_{l}$$_{−}$$_{1}$ log$^{(}$ p$_{l}$ ( z ) λ l − 1 i ( µ ) Δ$_{l}$$_{−}$$_{1}$ ) . (F1) ρ XABE ≡ 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ$_{0}$ ABE + | 1 〉〈 1 | X ⊗ ψ ABE 1 ] . l − 1 ∑ i =0 λ ( l − 1) i ( µ ) = Δ$_{l}$$_{−}$$_{1}$ , ∞ ∑ l =2 p$_{l}$ ( z ) = 1 . H ( B | X ) = H ( B ) − 1 + ∞ ∑ l =2 p$_{l}$ ( z ) log ( l − 1) − ∞ ∑ l =2 l − 1 ∑ i =0 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ λ ( l − 1) i ( µ ) log$^{(}$ λ ( l − 1) i ( µ )$^{)}$ . (F2) Appendix F: CQ and CE Trade-off Curves for the Unruh Channel N ( | 0 〉 〈 0 | ) ≡ ∞ ⊕ l =2 p$_{l}$ ( z ) S$_{l}$ ( | 0 〉 〈 0 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 ( l − 1 − i ) | i 〉〈 i | $^{B}$, N ( | 1 〉 〈 1 | ) ≡ ∞ ⊕ l =2 p$_{l}$ ( z ) S$_{l}$ ( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 ( i + 1) | i + 1 〉〈 i + 1 | $^{B}$. I ( X ; B ) = 1 − ∞ ∑ l =2 p$_{l}$ ( z ) log ( l − 1) + ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 1 ∑ i =0 λ ( l − 1) i ( µ ) log$^{(}$ λ ( l − 1) i ( µ )$^{)}$ . and we get the Holevo information: The following states are useful for calculating the Holevo information: The Holevo information I ( X ; B ) coincides with the ex- pression for the classical capacity of an Unruh channel when µ = 0 (Corollary 3, Equation (19) in Ref. [30], though note the slightly different definition of Δ$_{l}$ in that paper): The Holevo information vanishes when µ = 1 / 2. We now compute the coherent information. Let N $^{c}$( | 0 〉 〈 0 | ) and N $^{c}$( | 1 〉 〈 1 | ) denote the outputs of the complementary channel of the Unruh channel corre- sponding to the respective input states | 0 〉〈 0 | A ′ and I ( X ; B )$_{µ}$$_{=0}$ = 1 − ∞ ∑ l =2 p$_{l}$ ( z ) log ( l − 1)+ ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 1 ∑ i =0 i log i. ρ XB = 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ B 0 + | 1 〉〈 1 | X ⊗ ψ B 1 ] . ψ B 0 = µ N ( | 0 〉 〈 0 | ) + (1 − µ ) N ( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 1 ∑ i =0 λ l − 1 i ( µ ) | i 〉〈 i | $^{B}$, ψ B 1 = (1 − µ ) N ( | 0 〉 〈 0 | ) + µ N ( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 1 ∑ i =0 λ l − 1 i (1 − µ ) | i 〉〈 i | $^{B}$, ρ B = 1 2$^{(}$ N ( | 0 〉 〈 0 | ) + N ( | 1 〉 〈 1 | )$^{)}$ = ∞ ⊕ l =2 p$_{l}$ ( z ) l l − 1 ∑ i =0 | i 〉〈 i | $^{B}$,