The amplitude-damping channel has Kraus operators K_0= pmatrix 1&0 0& 1- pmatrix ,…

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Problem

The amplitude-damping channel has Kraus operators K0=(1001−γ),K1=γ ∣0⟩⟨1∣,K_0=\begin{pmatrix}1&0\\0&\sqrt{1-\gamma}\end{pmatrix},\qquad K_1=\sqrt\gamma\,|0\rangle\langle1|, for a two-level environment with basis ∣e0⟩,∣e1⟩|e_0\rangle,|e_1\rangle, starting in ∣e0⟩|e_0\rangle. ((a)) Write the isometry V∣ψ⟩=K0∣ψ⟩∣e0⟩+K1∣ψ⟩∣e1⟩V|\psi\rangle=K_0|\psi\rangle|e_0\rangle+K_1|\psi\rangle|e_1\rangle explicitly on ∣0⟩|0\rangle and ∣1⟩|1\rangle, and check directly that V†V=IV^\dagger V=I. ((b)) Complete VV to a 4×44\times4 unitary UU on the qubit and its environment, choosing the images of ∣0⟩∣e1⟩|0\rangle|e_1\rangle and ∣1⟩∣e1⟩|1\rangle|e_1\rangle so that all four columns are orthonormal. ((c)) For a general qubit state ρ\rho, check that tracing the environment out of U(ρ⊗∣e0⟩⟨e0∣)U†U\big(\rho\otimes|e_0\rangle\langle e_0|\big)U^\dagger returns K0ρK0†+K1ρK1†K_0\rho K_0^\dagger+K_1\rho K_1^\dagger, the amplitude-damping channel itself.

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