 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/quantum-information/2/d/ii/solution

Use

S = diag (1, i) = T^{2}

and the Pauli X gate. Starting with the Bell state

∣ Φ^{+} ⟩

, the exact constructions are

∣ Ψ_{1} ⟩ = (S \otimes X) ∣ Φ^{+} ⟩, ∣ Ψ_{2} ⟩ = (I \otimes H) ∣ Φ^{+} ⟩ .

For the first,

I \otimes X

changes the two terms to

∣01 ⟩

and

∣10 ⟩

, and

S \otimes I

multiplies just the second term by

i

. For the second,

(I \otimes H) ∣ Φ^{+} ⟩ = \frac{∣0 ⟩ ∣ + ⟩ + ∣1 ⟩ ∣ - ⟩}{2} = \frac{∣00 ⟩ + ∣01 ⟩ + ∣10 ⟩ - ∣11 ⟩}{2} .

All factors are single-party unitary operators, proving both claimed local-unitary equivalences. The equations hold as vectors, without even an additional global phase.

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