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

In quantum binary hypothesis testing, hypothesis zero supplies ${\rho }_0$ with prior $p$ and hypothesis one supplies ${\rho }_1$ with prior $1-p$. A two-outcome POVM $\{Q,I-Q\}$ decides zero on outcome $Q$. The conditional errors areSymmetric testing minimizes the prior-weighted average error $p\alpha +(1-p)\beta$, equivalently maximizing the average success probability.

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Let $\Delta =p{\rho }_0-(1-p){\rho }_1$. The success probability of $Q$ isWrite the spectral decomposition $\Delta ={\Delta }_{+}-{\Delta }_{-}$. For every effect $0\le Q\le I$,with equality when $Q$ projects onto the positive spectral subspace, with arbitrary action on the kernel. Since $\mathrm{Tr}\Delta =2p-1$ and $\Vert \Delta {\Vert }_1=\mathrm{Tr}{\Delta }_{+}+\mathrm{Tr}{\Delta }_{-}$, the Holevo–Helstrom theorem follows:

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