Let
$$
|\theta^\perp\rangle=\sin\theta|0\rangle-\cos\theta|1\rangle,
\qquad
c=\frac1{1+\cos\theta}.
$$
The three effects
$$
\boxed{E_1=c|\theta^\perp\rangle\langle\theta^\perp|},
\qquad
\boxed{E_2=c|1\rangle\langle1|},
\qquad
\boxed{E_?=I-E_1-E_2}
$$
form a <POVM>, because the largest eigenvalue of the sum of the two rank-one projectors is $1+\cos\theta$. Outcome 1 never occurs on $|\theta\rangle$, while outcome 2 never occurs on $|0\rangle$. Thus conclusive outcomes are never wrong; $E_?$ records failure. This is <unambiguous quantum state discrimination>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-323/5/b/i/solution