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 givesThus the $+$ outcome leaves $|{\psi }^{-}{⟩}_{23}$ and the $-$ outcome leaves $|{\psi }^{+}{⟩}_{23}$, 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: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 ${\mathrm{Tr}}_1|G⟩⟨G|$, as required by the no-communication theorem: a local quantum measurement with an unobserved outcome cannot change the remote reduced density matrix.