25 plify the above expression for the conditional entropy H ( E | X ): | 1 〉〈 1 | A $^{′}$: The following states are important in this calculation: We finally obtain the coherent information as the differ- ence of (F2) and (F3): H ( E | X ) = H ( B ) − 1 + ∞ ∑ l =2 p$_{l}$ ( z ) log ( l − 1) − ∞ ∑ l =2 l − 2 ∑ i =0 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ η ( l − 1) i ( µ ) log ( η ( l − 1) i ( µ )) . (F3) N $^{c}$( | 0 〉 〈 0 | ) ≡ ∞ ⊕ l =2 p$_{l}$ ( z ) S c $_{l}$( | 0 〉 〈 0 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 ( l − 1 − i ) | i 〉〈 i | E N $^{c}$( | 1 〉 〈 1 | ) ≡ ∞ ⊕ l =2 p$_{l}$ ( z ) S c $_{l}$( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 ( i + 1) | i 〉〈 i | $^{E}$. I ( A 〉 BX ) = − ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 1 ∑ i =0 λ ( l − 1) i ( µ ) log ( λ ( l − 1) i ( µ )) + ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 η ( l − 1) i ( µ ) log ( η ( l − 1) i ( µ )) . ρ XE = 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ E 0 + | 1 〉〈 1 | X ⊗ ψ E 1 ] , ψ E 0 = µ N $^{c}$( | 0 〉 〈 0 | ) + (1 − µ ) N $^{c}$( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 η ( l − 1) i ( µ ) | i 〉〈 i | $^{E}$, ψ E 1 = (1 − µ ) N $^{c}$( | 0 〉 〈 0 | ) + µ N $^{c}$( | 1 〉 〈 1 | ) = ∞ ⊕ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 η ( l − 1) i (1 − µ ) | i 〉〈 i | $^{E}$, I ( A 〉 BX )$_{µ}$$_{=1}$$_{/}$$_{2}$ = ∞ ∑ k =0 p$_{k}$$_{+2}$ ( z ) log$^{(}$ k + 2 k + 1$^{)}$ . The above expression vanishes when µ = 0, and it co- incides with the expression for the quantum capacity of the Unruh channel when µ = 1 / 2 (in Section III B of Ref. [33]): where η ( l − 1) i ( µ ) ≡ ( l − 2 − 2 i ) µ + i + 1. Then the condi- tional entropy H ( E | X ) is as follows: We now trace out the CE trade-off curve for a single use of the Unruh channel. We use the same input state as in Lemma 6 because that lemma proves that this input state traces out both the CQ curve and the CE curve. We then have here that H ( A | X ) = H$_{2}$ ( µ ), and we obtain the expression in the statement of the theorem by noting that I ( AX ; B ) = H ( A | X ) + H ( B ) − H ( E | X ). The relation ∑ l − 2 i $_{=0}$η ( l − 1) i ( µ ) = Δ$_{l}$$_{−}$$_{1}$ allows us to sim- H ( E | X ) = − ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 η ( l − 1)( µ ) i log$^{(}$ p$_{l}$ ( z ) η ( l − 1) i ( µ ) Δ$_{l}$$_{−}$$_{1}$ ) . March 2008. [8] Rochus Klesse. A random coding based proof for the quantum coding theorem. Open Systems & Information Dynamics , 15:21–45, March 2008. [1] Alexander S. Holevo. The capacity of the quantum chan- nel with general signal states. IEEE Transactions on Information Theory , 44(1):269–273, 1998. [2] Benjamin Schumacher and Michael D. Westmoreland. Sending classical information via noisy quantum chan- nels. Physical Review A , 56(1):131–138, July 1997. [3] Seth Lloyd. Capacity of the noisy quantum channel. Physical Review A , 55(3):1613–1622, March 1997. [4] Peter W. Shor. The quantum channel capacity and co- herent information. In Lecture Notes, MSRI Workshop on Quantum Computation , 2002. [5] Igor Devetak. The private classical capacity and quantum capacity of a quantum channel. IEEE Transactions on Information Theory , 51:44–55, January 2005. [6] Igor Devetak and Andreas Winter. Relating quantum pri- vacy and quantum coherence: An operational approach. Physical Review Letters , 93(8):080501, August 2004. [7] Patrick Hayden, Michal Horodecki, Andreas Winter, and Jon Yard. A decoupling approach to the quantum ca- pacity. Open Systems & Information Dynamics , 15:7–19, [12] Howard Barnum, Emmanuel Knill, and Michael A. Nielsen. On quantum fidelities and channel capacities. IEEE Transactions on Information Theory , 46(4):1317– 1329, July 2000. [11] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entan- glement and quantum error correction. Physical Review A , 54(5):3824–3851, November 1996. [10] Patrick Hayden, Peter W. Shor, and Andreas Winter. Random quantum codes from Gaussian ensembles and an uncertainty relation. Open Systems & Information Dynamics , 15:71–89, March 2008. [9] M. Horodecki, Seth Lloyd, and Andreas Winter. Quan- tum coding theorem from privacy and distinguishabil- ity. Open Systems & Information Dynamics , 15:47–69, March 2008.