7 probability and fidelity approaching unity in the limit of a large number of uses of the channel N . More precisely, the protocol transmits one classical message from a set of M messages and an arbitrary quantum state of di- mension K using a large number n uses of the quantum channel. The classical rate of transmission is C ≡ log( M ) n bits per channel use, and the quantum rate of transmis- sion is Q ≡ log( K ) n qubits per channel use. If there is a scheme that transmits classical data at rate C with van- ishing error probability and quantum data at rate Q with fidelity approaching unity in the limit of a large number of uses of the channel, we say that the rates C and Q form an achievable rate pair ( C, Q ). where | + 〉 and |−〉 are the eigenstates of the Pauli X matrix and | +$_{Y}$ 〉 and |−$_{Y}$ 〉 are the eigenstates of the Pauli Y matrix. The representation of the complemen- tary channel with unit rank Kraus operators explicitly demonstrates that it is entanglement-breaking (as first observed with a positive partial transpose argument in Ref. [30]), and the other arguments in Ref. [30] demon- strate that every 1 → N cloning channel is entanglement- breaking. Therefore, a 1 → N cloning channel is in the class of Hadamard channels. Devetak and Shor’s main result in Ref. [31] is that all achievable rate pairs lie in the following classically- enhanced quantum (CQ) capacity region of N : The Unruh channel is a natural channel to consider in the context of quantum field theory [35]. The output of the Unruh channel is the quantum state detected by a uniformly accelerated observer when the input is a dual- rail photonic qubit prepared by a Minkowski observer. The mathematical structure of the output of an Un- ruh channel N$_{U}$ is an infinite-dimensional block-diagonal density matrix, where the N th block is an instance of a 1 → N cloning channel: where Z is the closure of a set Z , and the “one-shot” region C $^{(1)}$( N ) is as follows: 5. Unruh channels C$_{CQ}$ ( N ) ≡ ∞ ⋃ k =1 1 k C (1) $_{CQ}$( N ⊗ $^{k}$) , (14) C (1) $_{CQ}$( N ) ≡ ⋃ ρ C (1) CQ, $_{ρ}$( N ) . N$_{U}$ ( σ ) ≡ ∞ ⊕ l =2 p$_{l}$ ( z ) S$_{l}$ ( σ ) , (13) p$_{l}$ ( z ) ≡ (1 − z ) $^{3}$z l − $^{2}$Δ$_{l}$$_{−}$$_{1}$ , C ≤ I ( X ; B )$_{ρ}$ , Q ≤ I ( A 〉 BX )$_{ρ}$ , The “one-shot, one-state” region C (1) CQ, $_{ρ}$( N ) is the set of all C, Q ≥ 0 such that the “acceleration parameter” z ∈ [0 , 1) is a strictly in- creasing function of acceleration, Δ$_{l}$$_{−}$$_{1}$ ≡ ( l − 1) l/ 2, and S$_{l}$ is the output of a 1 → ( l − 1) cloning channel. The complementary channel N c U is similar, with S$_{l}$ replaced by S c $_{l}$, the complementary channel of a 1 → ( l − 1) cloning channel. where where ρ is a classical-quantum state of the following form: the states φ AA ′ x are pure, and the dimension of the classi- cal system is finite [31]. For general channels, the multi- letter characterization in (14) is necessary, but for cer- tain channels, such as erasure channels and generalized dephasing channels, the CQ capacity region admits a single-letter characterization [31]. In Section IV A, we show that the CQ region for all Hadamard channels ad- mits a single-letter characterization. We review several trade-off capacity regions that have appeared in the quantum Shannon theory literature. We first review the Devetak-Shor result for the classically- enhanced quantum (CQ) capacity region [31]. We then review Shor’s results on the entanglement-assisted clas- sical (CE) capacity region [39], followed by a review of the natural generalization to the triple trade-off region for entanglement-assisted classical and quantum (CQE) coding [40]. ρ XABE ≡ ∑ x p$_{X}$ ( x ) | x 〉 〈 x | X ⊗ U A $^{′}$→ BE N ( φ AA ′ x ) , (15) B. Entanglement-Assisted Classical Capacity Region III. REVIEW OF CAPACITY REGIONS A. Classically-Enhanced Quantum Capacity Region Now consider a protocol that exploits shared noiseless entanglement and a noisy quantum channel N to trans- mit classical information. The goal of such a protocol is to transmit as much classical information as possible with vanishing error probability while consuming as little entanglement as possible in the limit of a large number of uses of the channel N . More precisely, the protocol transmits one classical message from a set of M messages Consider a protocol that exploits a noisy quantum channel N to transmit both classical and quantum in- formation. The goal of such a protocol is to transmit as much of these resources as possible with vanishing error