Past exam of the laser science and quantum information processing course of the University of Oxford / 2008 / 5 / 2 / Solution

The overlap $⟨0|+⟩=1/\sqrt{2}$ is nonzero, whereas perfect discrimination of pure states requires orthogonality. Consequently no single-copy quantum measurement can identify both preparations with certainty. After a nondegenerate projective measurement, the loss of preparation information after a nondegenerate projective measurement prevents further measurements from recovering the missing label: conditional on the first outcome, the post-measurement state is the same for every compatible initial hypothesis. Repeating the same basis gives that outcome again; changing basis probes the collapsed state. Fresh, independently prepared copies would be additional resources.

In the computational basis, input $|0⟩$ always gives zero and remains $|0⟩$. Input $|+⟩$ gives zero or one with probability $1/2$, leaving the corresponding basis state. Combining these likelihoods with equal prior probabilities, the Bayesian update after a projective measurement givesFor example, Bayes' theorem gives $P(0\text{ input}\mid 0\text{ outcome})=(1/2)/(3/4)=2/3$. Outcome one certifies input $|+⟩$; outcome zero is ambiguous. Both inputs that give outcome zero leave exactly the same post-measurement state $|0⟩$.

In the Hadamard basis, input $|+⟩$ always gives $+$, while input $|0⟩$ gives $+$ or $-$ equally. ThereforeOutcome minus certifies input $|0⟩$; outcome plus is ambiguous. Guessing the more likely input after the ambiguous outcome gives success probability $3/4$ for either of these two fixed-basis measurements.