6 decoherence process occurs naturally during stimulated emission [57–59]. We focus on 1 → N qubit cloning channels where a single qubit serves as the input, and the output is N identical approximate copies on N respective qubit sys- tems. These universal cloners are unitarily covariant [66], in the sense that a unitary V on the input qubit maps to an irreducible representation R$_{V}$ of V on the output state: channel is a Hadamard channel because its complemen- tary channel is entanglement-breaking. In fact, the iso- metric extension in (10) of the generalized dephasing channel appears remarkably similar to the isometric ex- tension in (8) of the Hadamard channel, with the excep- tion that the states {| i 〉 A $^{′}$} of the generalized dephasing channel form an orthonormal basis. A completely dephasing channel Δ commutes with a generalized dephasing channel N$_{GD}$ because The property N c GD = N c $_{GD}$◦ Δ also holds for the comple- mentary channel: We present the action of an isometric extension of a 1 → N cloning channel on a basis {| 0 〉 , | 1 〉} for a qubit input system A $^{′}$. Let | j 〉 B be an orthonormal basis of normalized completely symmetric states for the output system B that consists of N qubits: N ( V ρV ) = R$_{V}$ N ( ρ ) R † $_{V}$. ( N$_{GD}$ ◦ Δ)( σ ) = (Δ ◦ N$_{GD}$ )( σ ) = ∑ i 〈 i | σ | i 〉 A ′ | i 〉〈 i | $^{B}$. N c $_{GD}$( σ ) = ( N c GD ◦ Δ)( σ ) = ∑ i 〈 i | ρ | i 〉 A ′ | ϑ$_{i}$ 〉〈 ϑ$_{i}$ | $^{E}$. {| j 〉 B ≡ | N − j, j 〉} N j $_{=0}$, {| i 〉 E ≡ | N − i − 1 , i 〉} N − 1 i =0 , N ( σ ) = (1 − p ) σ + p Δ ( σ ) , (11) Δ( σ ) = 1 2 ( σ + ZσZ ) , U A $^{′}$→ BE N = √ 1 − p 2 I ⊗ | 0 〉 E +$^{√}$ p 2 Z ⊗ | 1 〉 $^{E}$, U A $^{′}$→ BE N$_{Cl}$ ≡ 1 √ Δ$_{N}$ N − 1 ∑ i =0 √ N − i | i 〉 $^{B}$〈 0 | A ′ ⊗ | i 〉 E + 1 √ Δ$_{N}$ N − 1 ∑ i =0 √ i + 1 | i + 1 〉 $^{B}$〈 1 | A ′ ⊗ | i 〉 $^{E}$, (12) where | N − j, j 〉 B denotes a normalized state on an N - qubit system that is a uniform superposition of compu- tational basis states with N − j “zeros” and j “ones.” Let | i 〉 E be an orthonormal basis for the environment E : The simplest example of a generalized dephasing chan- nel is a qubit dephasing channel. The action of the p - dephasing qubit channel is where Δ in this case is and Z is the Pauli matrix σ$_{Z}$ . Hence an isometric ex- tension U A $^{′}$→ BE N of the qubit dephasing channel has the form: where | N − i − 1 , i 〉 E denotes a normalized state on an ( N − 1)-qubit system that is a uniform superposition of computational basis states with N − i − 1 “zeros” and i “ones.” Then an isometric extension U A $^{′}$→ BE N$_{Cl}$ of the 1 → N cloning channel N has the form: where I is the identity operator. Therefore, the follow- ing is a complementary channel N c of a qubit dephasing channel: where Δ$_{N}$ ≡ N ( N + 1) / 2. A set of Kraus operators for the channel N$_{Cl}$ is as follows: and we observe that a bit flip on the input state does not change the eigenvalues of the resulting environment output state. and a set of Kraus operators for the complementary chan- nel N c Cl is as follows: { 1 √ Δ$_{N}$ $_{(}$√ N − i | i 〉 $^{B}$〈 0 | A ′ + √ i + 1 | i + 1 〉 $^{B}$〈 1 | A $^{′}$) } N − 1 i =0 , N $^{c}$( σ ) = p 2 | 0 〉 〈 0 | E + $^{(}$1 − p 2$^{)}$ | 1 〉 〈 1 | E +$^{√}$ $^{(}$1 − p 2$^{)}$ p 2 Tr { σZ } $^{(}$| 0 〉 〈 1 | E + | 1 〉 〈 0 | E $^{)}$, √ N | 0 〉 $^{E}$〈 0 | A ′ , $_{{}$√ N − i | i 〉 $^{E}$〈 0 | A ′ + √ i | i − 1 〉 $^{E}$〈 1 | A $^{′}$} N − 1 i =1 , √ N | N − 1 〉 $^{E}$〈 1 | A ′ .    √ 1 3 | + 〉 〈 + | , √ 1 3 |−〉 〈−| , √ 1 3 | 0 〉 〈 0 | , √ 1 3 | 1 〉 〈 1 | , √ 1 3 | +$_{Y}$ 〉 〈 +$_{Y}$ | σ$_{Z}$, √ 1 3 |−$_{Y}$ 〉 〈−$_{Y}$ | σ$_{Z}$    , We can rewrite the Kraus operators for the complemen- tary channel of a 1 → 2 cloning channel as follows: The no-cloning theorem forbids the cloning of arbitrary quantum states [64]. However, nothing prevents one from performing approximate cloning provided the fidelity of cloning is not too high. A universal 1 → N cloner is a device that approximately copies the input with maximal copy fidelity independent of the input state [65]. We refer to such a device as a cloning channel [30, 33]. Such a 4. Cloning Channels