User:Piotr gawron/Bhattacharyya angle

Definition
Bhattacharyya angle, also called angle or `statistical angle', is defined as

$$\Delta(p,q) = \mathrm{arccos} (BC(p,q))$$

for two $$N$$-point probability distributions it is given by the geodesic distance at the orthant of the sphere $$S^{N-1}$$ obtained by projecting the probability simplex on the sphere by the transformation $$p_i \rightarrow \sqrt{p_i},\ i=1\ldots N$$.

In the definition $BC(p,q)$ is Bhattacharyya coefficient defined as $$BC(p,q)=\sum_{i=1}^n \sqrt{p_i\cdot q_i}$$.

This distance is compatible with Fisher metric. It is also related to Bures distance and fidelity between quantum states as for two diagonal states one has

$$\Delta(\rho,\sigma) = \mathrm{arcos} \sqrt{F(\rho, \sigma)}.$$