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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.

Quantum state discrimination uses a measurement in quantum mechanics to identify which candidate density operator was prepared. Minimum-error and unambiguous discrimination impose different objectives; quantum binary hypothesis testing is the two-state case.