Pauli Matrices: A Triple of Accardi Complementary Observables

The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $su(2)$. We show that the pair of Pauli matrices $A,B$ associated to the unit directions $\alpha$ and $\beta$ in $\mathbb{R}^{3}$ are Accardi complementary if and only if $\alpha$ and $\beta$ are orthogonal if and only if $A$ and $B$ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary.