Quantum detection

Quantum detection infers a classical hypothesis from measurements of a quantum system. If hypothesis $j$ prepares a density matrix ${\rho }_j$, a measurement with POVM elements $E_k$ gives conditional probabilities $\mathrm{Tr}({\rho }_j{E}_k)$. Detector design chooses the measurement and decision rule for a stated error criterion.

For two equally probable states, the Holevo–Helstrom theorem gives minimum error $P_e=[1-\Vert {\rho }_0-{\rho }_1{\Vert }_1/2]/2$. Orthogonal states permit zero error; identical states force error $1/2$. The optimal binary measurement separates the positive and negative spectral parts of ${\rho }_0-{\rho }_1$, explaining why state geometry limits detection even with ideal apparatus.