16 that the sender would like to transmit classical data at a rate of 0 . 165 bits / channel use in order to transmit more quantum information. With a time-sharing strategy, the sender can transmit quantum data at approximately the rate 5 . 9 × 10 − 7 qubits / channel use, while classically- enhanced quantum coding transmits quantum data at approximately the rate 7 . 2 × 10 − 7 qubits / channel use. This improvement in transmission appears low in abso- lute terms, but the relative increase of transmission is substantial, and a measure that captures this relative in- crease may be more useful for studying the gain. We suggest an alternate gain measure that highlights the relative improvement and is simple to compute nu- merically for both the CQ and CE trade-off curves. Let A$_{CQ}$ denote the area under the CQ trade-off curve and let A$_{TSCQ}$ denote the area under the line of time-sharing. Then the relative gain G$_{CQ}$ for CQ trading is the ratio of A$_{CQ}$ to A$_{TSCQ}$ : Q$_{CQ}$ ( z, µ ) ≡ − ∞ ∑ 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 )) , C$_{CE}$ ( z, µ ) ≡ H$_{2}$ ( µ ) + 1 − ∞ ∑ l =2 p$_{l}$ ( z ) log ( l − 1) + ∞ ∑ l =2 p$_{l}$ ( z ) Δ$_{l}$$_{−}$$_{1}$ l − 2 ∑ i =0 η ( l − 1) i ( µ ) log ( η ( l − 1) i ( µ )) , E$_{CE}$ ( µ ) ≡ H$_{2}$ ( µ ) , λ ( l − 1) i ( µ ) ≡ ( l − 1 − 2 i ) µ + i, η ( l − 1) i ( µ ) ≡ ( l − 2 − 2 i ) µ + i + 1 , G$_{CQ}$ ≡ A$_{CQ}$ A$_{TSCQ}$ . G$_{CE}$ ≡ A$_{TSCE}$ A$_{CE}$ . and H$_{2}$ is the binary entropy function. Figure 4 plots both the CQ and CE trade-off curves of the Unruh channel for several values of the accelera- tion parameter z = 0, 0 . 2, 0 . 4, 0 . 6, 0 . 8, 0 . 95. The fig- ure demonstrates that both classically-enhanced quan- tum coding and entanglement-assisted classical coding beat a time-sharing strategy for an Unruh channel when z > 0. The relative gain measure for the CE trade-off curve is similar. Let A$_{CE}$ denote the area under the CE trade-off curve and let A$_{TSCE}$ denote the area under the line of time-sharing. Then the relative gain G$_{CE}$ for CE trading is the ratio of A$_{TSCQ}$ to A$_{CE}$ : Figure 5 plots the full CQE capacity region for an Unruh channel with acceleration parameter z = 0 . 95. The figure demonstrates that the classically-enhanced fa- ther protocol, combined with entanglement distribution and super-dense coding beats a time-sharing strategy be- cause the first two surfaces described in Section III C are strictly concave for the Unruh channel when z > 0. Each of the above relative gains exhibits non-trivial be- havior even if one of the capacities vanishes as the noise of a channel increases. These measures are also average gains because the area involves an integration over all points on the trade-off curve. One could generalize these relative gain measures to the CQE capacity region by taking the ratio of the volume enclosed by the bounding surfaces for CQE capacity region to the volume enclosed by surfaces obtained by time-sharing. Figures 1, 2, and 4 demonstrate that classically- enhanced quantum coding and entanglement-assisted classical coding both beat the time-sharing strategy for the dephasing, cloning, and Unruh channels. Ref. [31] provided a simple way to compute the benefit of “spe- cialized coding” over the time-sharing strategy, simply by plotting the difference between a trade-off curve and the line of time-sharing as a function of one of the rates in a trade-off scenario. Figure 6 plots the relative gains G$_{CQ}$ and G$_{CE}$ as a function of the dephasing parameter p for the dephasing qubit channel, as a function of the acceleration param- eter z for the Unruh channel, and as a function of the number of clones N for the 1 → N cloning channel. The accompanying caption features an interpretation of the results. One criterion that seems to be necessary in order to obtain a large relative gain for the CQ trade-off curve is that the quantum capacity of the channel should be much smaller than the classical capacity, so that the area between the inner bounding line of time-sharing and the trade-off curve is relatively large. The above gain measure illustrates the benefit of spe- cialized coding, but it ignores the relative improvement that specialized coding may give over time-sharing for very noisy channels. As a result, that gain measure tends to zero as one of the capacities tends to zero and thus loses meaning in this asymptotic limit. For exam- ple, consider a cloning channel with N = 1 , 000 , 000. This channel is particularly noisy for quantum trans- mission with a low quantum capacity at approximately 1 . 5 × 10 − 6 qubits / channel use, but the classical capac- ity is approximately 0 . 27 bits / channel use. Suppose We have proven that the CQE capacity region for all Hadamard channels admits a single-letter characteriza- tion. Particular examples of the Hadamard channels are VII. CONCLUSION VI. MEASURING THE GAIN OVER A TIME-SHARING STRATEGY