(b) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Classical communication rate Entanglement consumption rate Entanglement−assisted classical coding Time−sharin g (a) 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Quantum communication rate Classical communication rate Classically−enhanced quantum coding Time−sharin g 14 FIG. 2: (Color online) Plot of (a) the CQ trade-off curve and (b) the CE trade-off curve of a 1 → N cloning channel for N = 1, 2, 3, 5, 8, 12, and 24. The trade-off curves for N = 1 correspond to those for the noiseless qubit channel and are the rightmost trade-off curves in each plot. In both plots, proceeding left from the N = 1 curve, we obtain trade-off curves for N = 2, 3, 5, 8, 12, and 24 and notice that they all beat a time-sharing strategy by a larger relative proportion when N increases. giving the following expression for the Holevo informa- tion: The above expression coincides with the expression for the classical capacity of the 1 → N cloning channel when µ = 0 (Corollary 2 in Ref. [30]): The state at the output of the isometric extension is as follows: The expression in (31) vanishes when µ = 1 $_{2}$. We now compute the coherent information I ( A 〉 BX ) = H ( B | X ) − H ( E | X ). The following states are important in this computation: Defining λ$_{i}$ ( µ ) ≡ ( N − 2 i ) µ + i and Δ$_{N}$ = N ( N + 1) / 2 gives I ( X ; B ) = H ( B ) − H ( B | X ) = log ( N + 1) − H ( λ$_{i}$ ( µ ) / Δ$_{N}$) , (31) | ψ$_{1}$ 〉 ABE = 1 √ Δ$_{N}$$^{[}$$^{N}$$^{−}$$^{1}$ ∑ i =0 ( $^{√}$1 − µ √ N − i | 0 〉 $^{A}$| i 〉 B + √ µ √ i + 1 | 1 〉 $^{A}$| i + 1 〉 B ) | i 〉 E ] . ρ XABE ≡ 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ ABE 0 + | 1 〉〈 1 | X ⊗ ψ$_{1}$ ABE ] . I ( X ; B )$_{µ}$$_{=0}$ = 1 − log N + 1 Δ$_{N}$ N ∑ i =0 i log i ρ XE = 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ E 0 + | 1 〉〈 1 | X ⊗ ψ E 1 ] , ψ E 0 = N − 1 ∑ i =0 η$_{i}$ ( µ ) Δ$_{N}$ | i 〉〈 i | $^{E}$, (32) ψ E 1 = N − 1 ∑ i =0 η$_{i}$ ( µ ) Δ$_{N}$ | N − 1 − i 〉〈 N − 1 − i | $^{E}$, (33) ρ XB = 1 2$^{[}$ | 0 〉〈 0 | X ⊗ ψ B 0 + | 1 〉〈 1 | X ⊗ ψ B 1 ] , ψ B 0 = N ∑ i =0 λ$_{i}$ ( µ ) Δ$_{N}$ | i 〉〈 i | $^{B}$, ψ B 1 = N ∑ i =0 λ$_{i}$ ( µ ) Δ$_{N}$ | N − i 〉〈 N − i | $^{B}$, (29) ρ B = 1 N + 1 N ∑ i =0 | i 〉〈 i | $^{B}$. (30) H ( E | X ) = H ( η$_{i}$ ( µ ) / Δ$_{N}$) , I ( A 〉 BX ) = H ( λ$_{i}$ ( µ ) / Δ$_{N}$) − H ( η$_{i}$ ( µ ) / Δ$_{N}$) . H ( B ) = log ( N + 1) , H ( B | X ) = H ( λ$_{i}$ ( µ ) / Δ$_{N}$) , where η$_{i}$ ( µ ) ≡ ( N − 1 − 2 i ) µ + i +1. We can then compute the entropy H ( E | X ): and the coherent information is as follows: We can then compute the entropies H ( B ) and H ( B | X ):