 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2019/quantum-information/2/b/i/solution

The joint Born rule uses the tensor product of the two outcome bras. Its amplitude is

(⟨ 0∣ \otimes ⟨ θ_{0} ∣) ∣ ψ ⟩_{A B} = \frac{4}{5} ⟨ θ_{0} ∣0 ⟩ = \frac{4}{5} cos θ .

The other source component vanishes on contraction with

⟨ 0∣

on A. Therefore

Pr (A = 0, B = θ_{0}) = \frac{16}{25} cos^{2} θ .

This is a joint probability:

Pr (A = 0) = 16/25

, while

Pr (B = θ_{0} ∣ A = 0) = cos^{2} θ

. Dropping the first factor would give only the conditional probability.

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