 Pauli decomposition of a qubit-environment isometry

ID: pauli-decomposition-of-a-qubit-environment-isometry

Every linear map from a qubit into the tensor product of that qubit and an environment can be expanded in the Pauli matrix orthonormal basis of operators. If

V ∣0 ⟩ = ∣0 ⟩ ∣ e_{00} ⟩ + ∣1 ⟩ ∣ e_{01} ⟩

and

V ∣1 ⟩ = ∣0 ⟩ ∣ e_{10} ⟩ + ∣1 ⟩ ∣ e_{11} ⟩

, then the four coefficient vectors are

∣ e_{0} ⟩ = \frac{∣ e _{00} ⟩ + ∣ e _{11} ⟩}{2}, ∣ e_{1} ⟩ = \frac{∣ e _{01} ⟩ + ∣ e _{10} ⟩}{2}, ∣ e_{2} ⟩ = \frac{i ( ∣ e _{10} ⟩ - ∣ e _{01} ⟩)}{2}, ∣ e_{3} ⟩ = \frac{∣ e _{00} ⟩ - ∣ e _{11} ⟩}{2} .

These environment vectors need not be orthogonal, so this is an operator expansion rather than a claim that the quantum channel is a classical Pauli channel. Physical isometries additionally preserve the inner products of the two input basis states.

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