−0. 2 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 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 (a) 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 (b) 12 nel. Devetak and Shor gave a particular parametrization for the CQ trade-off curve for the case of a qubit dephas- ing channel in Appendix B of Ref. [31], and Hsieh and Wilde followed by giving a parametrization for the CE trade-off curve for qubit dephasing channels [40]. For the purposes of completion and comparison, we recall these parametrizations below, and Appendices D and E of this paper provide full proofs of these assertions. We now compute exact expressions for the CQ and CE trade-off curves, plot them for several 1 → N cloning channels, and plot the CQE capacity region for a par- ticular 1 → N cloning channel. The following theorem states these expressions, and the lemmas and subsequent calculation following it provide a proof. FIG. 1: (Color online) Plot of (a) the CQ trade-off curve and (b) the CE trade-off curve for a p -dephasing qubit channel for p = 0, 0 . 1, 0 . 2, . . . , 0 . 9, 1. The trade-off curves for p = 0 correspond to those of a noiseless qubit channel and are the rightmost trade-off curve in each plot. The trade-off curves for p = 1 correspond to those for a classical channel, and are the leftmost trade-off curves in each plot. Each trade-off curve between these two extremes beats a time-sharing strategy, but these two extremes do not beat time-sharing. B. 1 → N Cloning Channels Theorem 5 All points on the CQ trade-off curve for a 1 → N cloning channel have the following form: Theorem 4 All points on the CQ trade-off curve for a p -dephasing qubit channel have the following form: and all points on the CE trade-off curve for a p -dephasing qubit channel have the following form: where µ ∈ [0 , 1 / 2] , H$_{2}$ is the binary entropy function, and where H is the entropy function H ( · ) ≡ − $^{∑}$$_{i}$( · ) log ( · ) , Figure 1 plots both the CQ and CE trade-off curves for several p -dephasing qubit channels, where p varies from zero to one in increments of 1 / 10. The figure demon- strates that both classically-enhanced quantum coding and entanglement-assisted classical coding beat a time- sharing strategy for any p such that 0 < p < 1. We do not plot the full CQE capacity region for the dephasing qubit channel but instead point the interested reader to Figure 4 of Ref. [40]. Lemma 5 An ensemble of the following form parametrizes all points on the CQ trade-off curve Δ$_{N}$ ≡ N ( N + 1) / 2 , λ$_{i}$ ( µ ) ≡ ( N − 2 i ) µ + i for 0 ≤ i ≤ N, η$_{i}$ ( µ ) ≡ ( N − 1 − 2 i ) µ + i + 1 for 0 ≤ i ≤ N − 1 , µ ∈ [0 , 1 / 2] . ( log ( N + 1) + H$_{2}$ ( µ ) − H ( η$_{i}$ ( µ ) / Δ$_{N}$) , H$_{2}$ ( µ )$^{)}$ , ( log ( N + 1) − H$^{(}$ λ$_{i}$ ( µ ) Δ$_{N}$ ) , H$^{(}$ λ$_{i}$ ( µ ) Δ$_{N}$ ) − H$^{(}$ η$_{i}$ ( µ ) Δ$_{N}$ )) , (1 − H$_{2}$ ( µ ) , H$_{2}$ ( µ ) − H$_{2}$ ( γ ( µ, p ))) , (1 + H$_{2}$ ( µ ) − H$_{2}$ ( γ ( µ, p )) , H$_{2}$ ( µ )) , γ ( µ, p ) ≡ 1 2 + 1 2$^{√}$ 1 − 16 · p 2$^{(}$ 1 − p 2$^{)}$ µ (1 − µ ) . and all points on the CE trade-off curve for a 1 → N cloning channel have the following form: