3 isometric extension of the noisy map N , defined such that U † $_{N}$U$_{N}$ = I and in relativistic quantum information theory [30, 33, 35], bearing connections to the process of black-hole stimu- lated emission [60]. The proof technique to determine the formulas for the cloning channel extends naturally to the formulas for the Unruh channel, by exploiting the insights of Br´ adler in Ref. [30]. We also find that the coding strategy for each of these channels beats a sim- ple time-sharing strategy, and we introduce a measure to compute the amount by which it beats a time-sharing strategy. The Kraus operators provide a straightforward method for constructing an isometric extension: where the set {| i 〉 $^{E}$} is an orthonormal set of states. The following relation gives the conjugation of a density op- erator σ by the isometry U$_{N}$ : Let N c denote a complementary channel of N , unique up to isometries on the system E , whose action on σ is Observe that the following channel is a valid complemen- tary channel for the channel in (1): N A $^{′}$→ $^{B}$( σ ) = Tr$_{E}$ { U$_{N}$ σU † N } . U A $^{′}$→ BE N = ∑ i N A $^{′}$→ B i ⊗ | i 〉 E , U$_{N}$ σU † N = ∑ i,j ( N$_{i}$σN † $_{j}$) B ⊗ | i 〉 〈 j | E . (2) N $^{c}$( σ ) ≡ Tr$_{B}$ { U$_{N}$ σU † N } . N $^{c}$( σ ) = ∑ i,j Tr $^{{}$N$_{i}$ σN † j $^{}}$| i 〉 〈 j | E . We structure this work as follows. Section II reviews the definitions of noisy quantum channels, classical- quantum states, information-theoretic quantities, and examples of noisy quantum channels that are rele- vant for our purposes here. Section III then reviews the capacity regions mentioned above, specifically, the classically-enhanced quantum (CQ) capacity region, the entanglement-assisted classical (CE) capacity region, and the entanglement-assisted classical and quantum (CQE) capacity region—we abbreviate a capacity region by the noiseless resources involved: classical communication (C), quantum communication (Q), or entanglement (E). We then present our main result in Section IV: the proof that the CQ and CE capacity regions of Hadamard channels admit a single-letter characterization. As first observed by Hsieh and Wilde [40], and perhaps surprisingly, the single-letterization of these two regions immediately im- plies the single-letterization of the CQE capacity region. Section V computes the CQE capacity region of the qubit dephasing channel, the 1 → N cloning channel, and the Unruh channel, and the next section plots the CQE re- gions. The final section measures the improvement of the optimal protocols over a time-sharing strategy. Fi- nally, we conclude with some remaining observations and suggestions for future work. A quantum channel is degradable [31, 32], a notion im- ported from classical information theory [62], if there ex- ists a degrading channel D B → E that simulates the action of the complementary channel ( N $^{c}$) A $^{′}$→ E so that Suppose that Alice possesses an ensemble { ( p$_{X}$ ( x ) , ρ A ′ x ) } of quantum states where p$_{X}$ ( x ) is the probability density function for a random variable X and ρ A ′ x is a density operator conditional on the realization x of random variable X . She can augment this ensemble by correlating a classical variable with each ρ A ′ x . This procedure produces an augmented ensemble { ( p$_{X}$ ( x ) , | x 〉 〈 x | X ⊗ ρ A ′ x ) } , where the states {| x 〉 $^{X}$} form an orthonormal set. Taking the expecta- tion over the augmented ensemble gives the following classical-quantum state: We first review the notion of a noisy quantum channel, an isometric extension, and a classical-quantum state. A noisy quantum channel N is a completely-positive trace-preserving (CPTP) convex-linear map. It admits a Kraus representation [61], so that its action on a density operator σ is as follows: Let | φ$_{x}$ 〉 AA ′ denote purifications of each ρ A ′ x . The follow- ing state is also a classical-quantum state: where the Kraus operators form a resolution of the iden- tity: $^{∑}$$_{i}$N † $_{i}$N$_{i}$ = I , ensuring that the map is trace- preserving. The notion of an isometric extension of a noisy quantum channel proves to be useful in quantum Shannon theory—the notion is similar to that of a pu- rification of a density operator. Let U A $^{′}$→ BE N denote an Suppose that Alice transmits the A ′ subsystem through a noisy quantum channel N A $^{′}$→ $^{B}$. The state output from II. DEFINITIONS AND NOTATION A. Quantum Channels and Classical-Quantum States ∀ σ D B → E ◦ N A $^{′}$→ $^{B}$( σ ) = ( N $^{c}$) A $^{′}$→ E ( σ ) . ρ XA ′ ≡ ∑ x p$_{X}$ ( x ) | x 〉〈 x | X ⊗ ρ A ′ x . ρ XAA ′ ≡ ∑ x p$_{X}$ ( x ) | x 〉〈 x | X ⊗ | φ$_{x}$ 〉〈 φ$_{x}$ | AA $^{′}$. (3) N ( σ ) = ∑ i N$_{i}$σN † $_{i}$, (1)