Consider a two-dimensional system interacting with a qubit environment via the unitary produced by the circuit, where the system is the top qubit labelled S and the environment is the bottom qubit labelled E. The one qubit gates are the Pauli matrices and .

a) Compute the matrix of with respect to the standard basis in ^{2 2} .

b) Compute the action of on the input state | |0 where | = |0 + |1.

c) Consider the quantum channel describing the evolution of the first qubit, when the second qubit is initially in the state |0

Compute () for = ||, with | = |0 + |1.

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