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.