5 in the following commutative diagram: rank and the construction in (2) for an isometric exten- sion, we can write an isometric extension U$_{N}$ c for N c as The sets {| ξ$_{i}$ 〉 $^{E}$} and {| ζ$_{i}$ 〉 A $^{′}$} each do not necessarily con- sist of orthonormal states, but the set {| i 〉 $^{B}$} does. Trac- ing over the system E gives the original channel from system A ′ to B : We show in the next two sections that generalized de- phasing channels and cloning channels are both mem- bers of the Hadamard class because their complementary channels are entanglement-breaking. B A ′ N H ✲ Y (classical) D$_{1}$ ❄ . . . . . E D$_{2}$ ❄ . . . . N c H ✲ U$_{N}$ c σU † N c = ∑ i,j | ξ$_{i}$ 〉 E 〈 ζ$_{i}$ | A ′ σ | ζ$_{j}$ 〉 A ′ 〈 ξ$_{j}$ | E ⊗ | i 〉 B 〈 j | B = ∑ i,j 〈 ζ$_{i}$ | A ′ σ | ζ$_{j}$ 〉 A ′ | ξ$_{i}$ 〉 E 〈 ξ$_{j}$ | E ⊗ | i 〉 B 〈 j | B . (8) N A $^{′}$→ B H ( σ ) = ∑ i,j 〈 ζ$_{i}$ | A ′ σ | ζ$_{j}$ 〉 A ′ 〈 ξ$_{j}$ | ξ$_{i}$ 〉 E | i 〉 B 〈 j | B . (9) Generalized dephasing channels model physical pro- cesses where there is no loss of energy but there is a loss of quantum coherence [61], a type of quantum noise that dominates for example in superconducting qubits [56]. The respective input and output systems A ′ and B of such channels are of the same dimension. Let {| i 〉 A $^{′}$} and {| i 〉 $^{B}$} be some respective preferred orthonormal bases for these systems, the first of which we call the dephas- ing basis. The channel does not affect any state that is diagonal in the dephasing basis, but it mixes coherent superpositions of these basis states. Let Σ denote the matrix with elements [Σ]$_{i,j}$ = 〈 ζ$_{i}$ | A ′ σ | ζ$_{j}$ 〉 A ′ , a representation of the input state σ , and let Γ denote the matrix with elements [Γ]$_{i,j}$ = 〈 ξ$_{j}$ | ξ$_{i}$ 〉 $^{E}$. Then, from (9), it is clear that the output of the channel is the Hadamard product ∗ of Σ and Γ † with respect to the basis {| i 〉 $^{B}$} : For this reason, such a channel is known as a Hadamard channel. Hadamard channels are degradable. If Bob per- forms a von Neumann measurement of his state in the basis {| i 〉 $^{B}$} and prepares the state | ξ$_{i}$ 〉 E conditional on the outcome of the measurement, this procedure sim- ulates the complementary channel ( N $^{c}$) A $^{′}$→ E and also implies that the degrading map D B → E is entanglement- breaking. To be more precise, the Kraus operators of the degrading map D B → E are {| ξ$_{i}$ 〉 E 〈 i | $^{B}$} so that An isometric extension U A $^{′}$→ BE N$_{GD}$ of a generalized de- phasing channel N A $^{′}$→ B GD has the form: where the set {| ϑ$_{i}$ 〉 $^{E}$} is not necessarily an orthonormal set. The output of a generalized dephasing channel is as follows: 3. Generalized Dephasing Channels U A $^{′}$→ BE N$_{GD}$ ≡ ∑ i | i 〉 B 〈 i | A ′ ⊗ | ϑ$_{i}$ 〉 E , (10) N A $^{′}$→ B H ( σ ) = Σ ∗ Γ $^{†}$. N$_{GD}$ ( σ ) = ∑ i,j 〈 ϑ$_{j}$ | ϑ$_{i}$ 〉 E 〈 i | σ | j 〉 A ′ | i 〉〈 j | $^{B}$. D B → $^{E}$( N A $^{′}$→ B H ( σ )) = ∑ i | ξ$_{i}$ 〉 E 〈 i | B N A $^{′}$→ B ( σ ) | i 〉 B 〈 ξ$_{i}$ | E = ∑ i 〈 i | A ′ σ | i 〉 A ′ | ξ$_{i}$ 〉 〈 ξ$_{i}$ | E , Δ A $^{′}$→ $^{B}$( σ ) ≡ ∑ i | i 〉 $^{B}$〈 i | A $^{′}$σ | i 〉 A $^{′}$〈 i | $^{B}$. If the states | ϑ$_{i}$ 〉 E are orthonormal, the channel is a com- pletely dephasing channel Δ A $^{′}$→ $^{B}$: We obtain the complementary channel of a generalized dephasing channel by tracing over Bob’s system B : demonstrating that this degrading map effectively simu- lates the complementary channel ( N c $_{H}$) A $^{′}$→ $^{E}$. Note that we can view this degrading map as the composition of two maps: a first map D B → Y 1 performs the von Neumann measurement, leading to a classical variable Y , and a sec- ond map D Y → E 2 performs the state preparation, condi- tional on the value of the classical variable Y . We can therefore write D B → E = D Y → E 2 ◦ D B → Y 1 . This obser- vation is crucial to our proof of the single-letterization of both the CQ and CE capacity regions of Hadamard channels. These structural relationships are summarized which we recognize to be an entanglement-breaking chan- nel of the form in (7). Thus, a generalized dephasing Tr$_{B}$ $^{{}$U$_{N}$$_{GD}$ σU † N$_{GD}$ } = ∑ i 〈 i | A ′ σ | i 〉 A ′ | ϑ$_{i}$ 〉 〈 ϑ$_{i}$ | E ,