= Uniform pure-qubit average
{title2=$d\mu=\frac{\sin\theta\,d\theta\,d\varphi}{4\pi}$}
= Haar-uniform qubit average
{c}
{synonym}
The rotation-invariant average of pure <qubit> states is uniform area measure on the <Bloch sphere>, induced by <Haar measure> on unitary transformations. For a uniformly distributed unit <Bloch vector> $r$, rotational symmetry gives $\mathbb Er_i=0$ and $\mathbb Er_ir_j=\delta_{ij}/3$. Uniform polar angle without the factor $\sin\theta$ is a different distribution.
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