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

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Past exam of the short options paper of the physics Prelims of the University of Oxford / 2022 / S03 / 3 / d / Solution by

Codex  0 2026-10-25

Expand a normalized product state as

(a ∣0 ⟩ + b ∣1 ⟩) \otimes (c ∣0 ⟩ + d ∣1 ⟩) = a c ∣00 ⟩ + a d ∣01 ⟩ + b c ∣10 ⟩ + b d ∣11 ⟩ .

Thus

α = a c

,

β = a d

,

γ = b c

,

δ = b d

α δ = ab c d = β γ .

Conversely the nonzero coefficient matrix

C = (α γ β δ)

has determinant zero precisely when it has rank one, so the condition also suffices for a pure two-qubit state to factor.

For any fixed pair of local measurement settings with operators

E_{i}, F_{j}

, the Born rule factorizes:

p (i, j) = ⟨ a ∣ E_{i} ∣ a ⟩ ⟨ b ∣ F_{j} ∣ b ⟩ = p_{A} (i) p_{B} (j) .

Hence a product state has independent local results in every pair of local bases. This assumes independent settings and local measurements; shared classical randomization of preparations or a subsequent joint interaction can generate correlations. A separable mixture is not necessarily a single product state.

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