# Pauli Matrices: A Triple of Accardi Complementary Observables

## TL;DR

This paper adapts Accardi's definition of complementary observables to the Lie algebra su(2), showing that pairs of Pauli matrices are complementary if and only if their associated directions are orthogonal.

## Contribution

It establishes a precise geometric condition for Pauli matrices to be Accardi complementary within the su(2) Lie algebra.

## Key findings

- Pauli matrices are Accardi complementary iff their directions are orthogonal.
- Any pair of standard Pauli matrices is complementary.
- Complementarity is characterized by orthogonality of directions.

## Abstract

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.