 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-35/4/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

False. Take the Bell state

∣ Φ^{+} ⟩ = \frac{∣00 ⟩ + ∣11 ⟩}{2}, ρ = ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣.

The joint state is pure, so

S (ρ) = 0

. Taking either partial trace kills the off-diagonal terms and gives

ρ_{1} = ρ_{2} = I_{2} /2

, whose two eigenvalues are

1/2

. Thus

S (ρ_{1}) = S (ρ_{2}) = 1

, contradicting the proposed inequality

1 \leq 0

. Quantum entanglement allows a pure joint state to have mixed subsystem states.

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