Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 51 / 1 / d / Solution

In the ordered computational basis $00,01,10,11$, the first two qubits of $|{\psi }_{0000}⟩$ have amplitudes $(1,1,1,1)/2$, while those of $|{\psi }_{1111}⟩$ have amplitudes $(-1,1,1,1)/2$. Both have the same normalized answer qubit $|-⟩$. Their inner product is thereforeA common unitary operator $U\otimes I$ preserves this nonzero inner product. To distinguish the alternatives with certainty using the prescribed quantum measurement in the computational basis, their first-register outcome supports would have to be disjoint: every outcome possible under one alternative must be impossible under the other. Vectors with disjoint computational basis supports have zero inner product, contradicting its preservation. Thus no such $U$ exists. This is also an instance of perfect discrimination of pure states requires orthogonality; allowing a more general measurement in quantum mechanics would not rescue this pair of states. The obstruction concerns these prepared oracle-output states, rather than every conceivable oracle-query procedure.