Let $\Delta=p\rho_0-(1-p)\rho_1$. The success probability of $Q$ is
$$
P_{\rm succ}(Q)
=p\operatorname{Tr}(Q\rho_0)
+(1-p)\operatorname{Tr}[(I-Q)\rho_1]
=1-p+\operatorname{Tr}(Q\Delta).
$$
Write the spectral decomposition $\Delta=\Delta_+-\Delta_-$. For every effect $0\leq Q\leq I$,
$$
\operatorname{Tr}(Q\Delta)
\leq\operatorname{Tr}\Delta_+,
$$
with equality when $Q$ projects onto the positive spectral subspace, with arbitrary action on the kernel. Since $\operatorname{Tr}\Delta=2p-1$ and $\|\Delta\|_1=\operatorname{Tr}\Delta_++\operatorname{Tr}\Delta_-$, the <Holevo–Helstrom theorem> follows:
$$
\boxed{P_{\rm succ}^*
=\frac12(1+\|\Delta\|_1)},
\qquad
\boxed{P_{\rm err}^*
=\frac12(1-\|\Delta\|_1)}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 3 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-323/3/a/ii/solution