 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/quantum-information/1/a/solution

For the three given qutrit vectors,

⟨ ϕ_{0} ∣ = (⟨ 0∣ - i ⟨ 2∣) / 2

and

⟨ ϕ_{2} ∣ = (⟨ 0∣ + i ⟨ 2∣) / 2

. Taking the complex conjugate in the bra is essential. Their squared norms are

1, 1, 1

, and

⟨ ϕ_{0} ∣ ϕ_{2} ⟩ = \frac{1}{2} {1 + (- i) (- i)} = 0, ⟨ ϕ_{0} ∣ ϕ_{1} ⟩ = ⟨ ϕ_{2} ∣ ϕ_{1} ⟩ = 0.

Thus they are an orthonormal set of three vectors in the three-dimensional inner product space

C^{3}

. Such a set is linearly independent and spans the space, so it is an orthonormal basis.

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