= 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 $\operatorname{Tr}(\rho_jE_k)$. 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 $P_e=[1-\|\rho_0-\rho_1\|_1/2]/2$. 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.
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