 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2022/s03/3/b/solution

Measuring the first qubit applies

P_{0} = ∣0 ⟩ ⟨ 0∣ \otimes I

P_{1} = ∣1 ⟩ ⟨ 1∣ \otimes I

. The Born rule gives

p_{0} = ∣ α ∣^{2} + ∣ β ∣^{2}, p_{1} = ∣ γ ∣^{2} + ∣ δ ∣^{2} = 1 - p_{0} .

The normalized conditional quantum states are

∣ ψ_{0} ⟩ = \frac{α ∣00 ⟩ + β ∣01 ⟩}{p _{0}}, ∣ ψ_{1} ⟩ = \frac{γ ∣10 ⟩ + δ ∣11 ⟩}{p _{1}} .

Each expression applies only when its denominator is nonzero; an impossible outcome has no defined conditional state.

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