Bhattacharyya coefficient ($B(p,q)={\sum }_i\sqrt{p_i{q}_i}$)

For probability distributions $p,q$ on the same finite alphabet, the Bhattacharyya coefficient measures their square-root overlap:The vectors $(\sqrt{p_i}{)}_i$ and $(\sqrt{q_i}{)}_i$ have unit Euclidean norm, so the Cauchy-Schwarz inequality gives $0\le B\le 1$. Equality $B=1$ holds exactly when the distributions agree; $B=0$ holds exactly when they assign positive probability to disjoint outcomes. For $p=(0.9,0.1)$ and $q=(0.6,0.4)$, $B=\sqrt{0.54}+0.2\approx 0.934847$. This overlap is not the probability that two independent draws agree, which here is $0.58$.

The fidelity of commuting mixed states reduces to this coefficient. More generally, the measurement characterization of quantum fidelity minimizes it over outcome distributions of a common measurement in quantum mechanics, connecting classical overlap to the unsquared quantum fidelity.

The Bhattacharyya distance is the negative natural logarithm of the Bhattacharyya coefficient, with $-\ln 0=\infty$. It is zero for identical probability distributions and infinite for disjoint probability supports. Despite the name, it need not satisfy the triangle inequality required of a metric.