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

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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