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3. (5pts) (Exercise 8.4 from textbook) Suppose that we have a system comprised of a single qubit interacting with an environment comprised also of a single qubit. The interaction between system and environment is given by the unitary operation \[ \hat{U}=|0\rangle\langle 0|\otimes \hat{I}+| 1\rangle\langle 1| \otimes \hat{X}, \] where \( \hat{I} \) is the identity and \( \hat{X} \) is the usual Pauli matrix that both act on the environment (this equivalent to a CNOT interaction controlled by the system and acting on the environment). Give the quantum operation for this process, in the operator sum representation, assuming that the environment starts in the state \( |0\rangle \) and we then lose all information about the environment after it interacts with the system (i.e. trace our the environment).

          3. (5pts) (Exercise 8.4 from textbook) Suppose that we have a system comprised of a single qubit interacting with an environment comprised also of a single qubit. The interaction between system and environment is given by the unitary operation
\[
\hat{U}=|0\rangle\langle 0|\otimes \hat{I}+| 1\rangle\langle 1| \otimes \hat{X},
\]
where \( \hat{I} \) is the identity and \( \hat{X} \) is the usual Pauli matrix that both act on the environment (this equivalent to a CNOT interaction controlled by the system and acting on the environment). Give the quantum operation for this process, in the operator sum representation, assuming that the environment starts in the state \( |0\rangle \) and we then lose all information about the environment after it interacts with the system (i.e. trace our the environment).

3. (5pts) (Exercise 8.4 from textbook) Suppose that we have a system comprised of a single qubit interacting with an environment comprised also of a single qubit. The interaction between system and environment is given by the unitary operation

Û=|0⟩⟨ 0|⊗Î+| 1⟩⟨ 1| ⊗X̂,

where Î is the identity and X̂ is the usual Pauli matrix that both act on the environment (this equivalent to a CNOT interaction controlled by the system and acting on the environment). Give the quantum operation for this process, in the operator sum representation, assuming that the environment starts in the state |0⟩ and we then lose all information about the environment after it interacts with the system (i.e. trace our the environment).

Added by Javier P.

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