18 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.05 1.1 1.15 p, z Relative improvement over time−sharing 1 100 200  300 400  500 600  700 800  900 1000 Number of Clones Cloning Channel Dephasing Channel Unruh 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.005 1.01 1.015 1.02 1.025 1.03 1.035 1.04 p, z Relative Improvement over time−sharing 1 100 200 300 400 500 600 700 800 900 1000 Number of Clones Cloning Channel Dephasing Channel Unruh (b) (a) FIG. 6: (Color online) The figures plot (a) the relative gain G$_{CQ}$ for the CQ trade-off curve and (b) the relative gain G$_{CE}$ for the CE trade-off curve for the dephasing channel, the cloning channel, and the Unruh channel. The figures plot these relative gains as a function of the dephasing parameter p ∈ [0 , 1] for the p -dephasing qubit channel (bottom horizontal axis), as a function of the acceleration parameter z ∈ [0 , 1] for the Unruh channel (bottom horizontal axis), and as a function of the number of clones N for the 1 → N cloning channel (top horizontal axis). The plot on the left demonstrates that the relative gain G$_{CQ}$ for the cloning channels is best as N increases, while the Unruh channel features an improved relative gain over a dephasing channel for large accelerations. The plot on the right features different behavior—the relative gain G$_{CE}$ of the dephasing channel outperforms that for the Unruh channel if we consider the parameters p and z on equal footing, in spite of their drastically different physical interpretations. sider the CQE capacity region for a qudit Unruh channel by exploiting some of the insights in Ref. [67]. Finally, one might consider a trade-off capacity region for an Un- ruh channel that includes the resource of private quan- tum information, given that this noiseless resource ap- pears in relativistic quantum information theory, or one could consider trading public classical information, pri- vate classical information, and secret key by exploiting the ideas in Refs. [68, 69]. likely that the technique for proving single-letterization would be completely different. The ideas exploited here are that Hadamard channels are degradable and have entanglement-breaking complementary channels, allow- ing us to generalize the Devetak-Shor proof technique in Appendix B of Ref. [31]. If other single-letter examples do exist, one should then determine if such a channel obtains an improvement over time-sharing and perhaps attempt to uncover a general method that determines if a channel obtains a gain over time-sharing. It may be interesting to explore the static case, where two parties share a bipartite state and exploit this state and some noiseless resources to extract other noiseless resources. One might consider bipartite states that arise from Hadamard channels and attempt to single-letterize the static capacity region. Hsieh and Wilde found for- mulas for the triple trade-off static capacity region in Ref. [41], but the task of single-letterization for the static case is more difficult than that for the dynamic case be- cause one must consider quantum instruments applied to many copies of the bipartite state. It would also be interesting to consider versions of the Unruh channel other than the original definition in Ref. [35] and determine if the corresponding CQE capac- ity region can single-letterize (we refer to this channel as “the” Unruh channel, but it may be more appropriate to consider it as “an” Unruh channel). One can con- The authors acknowledge useful discussions with Min- Hsiu Hsieh concerning the CQ and CE trade-off curves and the CQE capacity region. K. B. and P. H. ac- knowledge support from the Office of Naval Research under grant No. N000140811249. P. H. acknowledges support from the Canada Research Chairs program, CI- FAR, FQRNT, MITACS, NSERC, QuantumWorks, and the Sloan Foundation. M. M. W. acknowledges support from the MDEIE (Qu´ ebec) PSR-SIIRI international col- laboration grant. Acknowledgments