Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 323 / 5 / b / i / Solution

LetThe three effectsform a POVM, because the largest eigenvalue of the sum of the two rank-one projectors is $1+\cos \theta$. Outcome 1 never occurs on $|\theta ⟩$, while outcome 2 never occurs on $|0⟩$. Thus conclusive outcomes are never wrong; $E_{?}$ records failure. This is unambiguous quantum state discrimination.