2 classically-enhanced quantum capacity region (though the latter does imply the former). That is, the proof that the region single-letterizes requires a different tech- nique that has no obvious connection to the techniques used to prove the single-letterization of the individual capacities. The additional benefit of the Devetak-Shor classically-enhanced quantum coding scheme is that it beats a time-sharing strategy [72] for some channels. nels [27], transpose-depolarizing channels [28], shifted de- polarizing channels [29], cloning channels [30], and the so-called “Unruh” channel [30]. Researchers have found fewer such exemplary single-letter gems for the quan- tum capacity: erasure channels [24], degradable chan- nels [31, 32], conjugate degradable channels [33], and amplitude damping channels [34]. We do not yet have a general method for determining whether a given chan- nel’s capacity admits a single-letter characterization us- ing known formulas—the techniques for proving single- letterization of the above examples are all ad hoc , vary- ing from case to case. Additionally, we can observe that it is an even rarer gem for a channel to admit a single-letter characterization for both the classical and quantum capacity. This “single-letter overlap” occurs for erasure channels [24], generalized dephasing chan- nels [31], cloning channels [30, 33], and the Unruh chan- nel [30, 33, 35], but the reasons for single-letterization of the classical and quantum capacities have no obvious connection. We might say that the above scenarios are a part of the dynamic, double-resource quantum Shannon theory, where a sender can exploit a noisy quantum channel to generate two noiseless resources, or a sender can exploit a noisy quantum channel in addition to a noiseless re- source to generate another noiseless resource. This the- ory culminated with the landmark work of Devetak et al . that provided a multi-letter characterization for vir- tually every permutation of two resources and a noisy quantum channel which one can consider [44, 45], but they neglected to search for channels with single-letter characterizations of the double-resource capacity regions. Other researchers concurrently considered how noiseless resources might trade off against each other in tasks out- side of the dynamic, double-resource quantum Shannon theory, such as quantum compression [46–48], remote state preparation [49, 50], and hybrid quantum memo- ries [51]. After the early work in quantum Shannon theory, sev- eral researchers considered how different noiseless re- sources such as entanglement, classical communication, or quantum communication might trade off against one another together with a noisy channel. The first find- ings in this direction were those of Bennett et al . [36, 37], who showed that unlimited, shared, noiseless entangle- ment can boost the classical capacity of a noisy quan- tum channel, generalizing the super-dense coding effect [38]. Perhaps even more surprising was that the formula for the entanglement-assisted classical capacity gives a single-letter characterization, marking the first time that we could say there is a problem in quantum Shannon theory that we truly understand. Shor then refined this result by considering the classical capacity of a channel assisted by a finite amount of shared entanglement [39]. He calculated a trade-off curve that determines how a sender can optimally trade the consumption of noiseless entanglement with the generation of noiseless classical communication. This trade-off curve also bounds a rate region consisting of rates of entanglement consumption and generated classical communication. Unfortunately, the formulas for the rate region do not give a single-letter characterization in the general case. The next natural step in this line of inquiry was to consider the dynamic, triple-resource quantum Shannon theory. Hsieh and Wilde did so by providing a multi- letter characterization of an entanglement-assisted quan- tum channel’s ability to transmit both classical and quan- tum information [40, 41]. In addition, they found that the formulas for a generalized dephasing channel, an erasure channel, and the trivial completely depolarizing chan- nel all give a single-letter triple trade-off capacity region (though they omitted the full proof for the generalized de- phasing channels). Thus, these findings provided a com- plete understanding of the ultimate performance limits of any scheme for entanglement-assisted classical and quan- tum error correction [42, 43, 52–55], at least for the afore- mentioned channels. Ref. [40] also constructed a new protocol, dubbed the “classically-enhanced father proto- col,” that outperforms a time-sharing strategy for trans- mitting both classical and quantum information over an entanglement-assisted quantum channel. Shor’s result then inspired Devetak and Shor to con- sider a scenario where a sender exploits a noisy quan- tum channel to simultaneously transmit both noiseless classical and quantum information [31], a scenario later dubbed “classically-enhanced quantum coding” [40, 41] after schemes formulated in the theory of quantum error correction [42, 43]. Devetak and Shor provided a multi- letter characterization of the classically-enhanced quan- tum capacity region for general channels, but were able to show that both generalized dephasing channels and erasure channels admit single-letter capacity regions. We must emphasize that single-letterization of both the clas- sical and quantum capacities of a noisy quantum channel does not immediately imply the single-letterization of the In this paper, we contribute a class of “single-letter gems” to the dynamic, triple-resource quantum Shan- non theory. We provide single-letter formulas for the triple trade-off capacity region of the Hadamard class of channels. We then compute and plot examples of the triple trade-off regions for the generalized dephas- ing channels, the cloning channels, and the Unruh chan- nel, all of which are members of the class of Hadamard channels. The generalized dephasing channel represents a natural mechanism for decoherence in physical systems such as superconducting qubits [56], the cloning chan- nel represents a natural process that occurs during stim- ulated emission [57–59], and the Unruh channel arises