 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2005/7-and-8/8/3/solution

Keep the actual odd-

n

GHZ state from the preceding calculation. Resolving its first qubit in the Hadamard basis gives

∣ G ⟩ = \frac{∣ + ⟩ _{1} ∣ ψ ^{-} ⟩ _{23} + ∣ - ⟩ _{1} ∣ ψ ^{+} ⟩ _{23}}{2}, ∣ ψ^{\pm} ⟩_{23} = \frac{∣01 ⟩ \pm ∣10 ⟩}{2} .

Thus the

+

outcome leaves

∣ ψ^{-} ⟩_{23}

and the

-

outcome leaves

∣ ψ^{+} ⟩_{23}

, each with probability

1/2

. Conditional on a known result the remaining qubits share a maximally entangled state.

When the result is not revealed, the appropriate density matrix is the probability-weighted mixture from a nonselective quantum measurement:

ρ_{23} = \frac{1}{2} ∣ ψ^{-} ⟩ ⟨ ψ^{-} ∣ + \frac{1}{2} ∣ ψ^{+} ⟩ ⟨ ψ^{+} ∣ = \frac{1}{2} ∣01 ⟩ ⟨ 01∣ + \frac{1}{2} ∣10 ⟩ ⟨ 10∣ = \frac{1}{2} (∣0 ⟩ ⟨ 0∣ \otimes ∣1 ⟩ ⟨ 1∣) + \frac{1}{2} (∣1 ⟩ ⟨ 1∣ \otimes ∣0 ⟩ ⟨ 0∣) = diag (0, 1/2, 1/2, 0) .

The cross terms cancel. The displayed convex combination of product states proves directly that this is a separable quantum state. It still has perfect computational-basis measurement anticorrelation, so it is not itself a product state. It also equals the partial trace

Tr_{1} ∣ G ⟩ ⟨ G ∣

, as required by the no-communication theorem: a local quantum measurement with an unobserved outcome cannot change the remote reduced density matrix.

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