= Bhattacharyya coefficient
{c}
{title2=$B(p,q)=\sum_i\sqrt{p_iq_i}$}
For <probability distributions> $p,q$ on the same finite alphabet, the Bhattacharyya coefficient measures their square-root overlap:
$$
B(p,q)=\sum_i\sqrt{p_iq_i}.
$$
The vectors $(\sqrt{p_i})_i$ and $(\sqrt{q_i})_i$ have unit <Euclidean norm>, so the <Cauchy-Schwarz inequality> gives $0\leq B\leq1$. 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 $q=(0.6,0.4)$, $B=\sqrt{0.54}+0.2\approx0.934847$. 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 <quantum measurement>, connecting classical overlap to the <unsquared quantum fidelity>.
Back to article page
