Contents

group theory

# Contents

## Definition

The Pauli group $\mathcal{P}_1$ is the finite group of order 16 which is isomorphic to the subgroup of the complex general linear group $GL(2,\mathbb{C})$ that consists of the multiples by $\pm 1$ and $\pm i$ of the four Pauli matrices $\sigma_{0,1,2,3}$:

(1)$\mathcal{P}_1 \;\coloneqq\; \big\{ \pm \sigma_0, \, \pm \sigma_1, \, \pm \sigma_2, \, \pm \sigma_3, \,\; \pm i \sigma_0, \, \pm i \sigma_1, \, \pm i \sigma_2, \, \pm i \sigma_3, \big\} \;\subset\; GL(2,\mathbb{C}) \,,$

where

$\sigma_0 \;\coloneqq\; \left[ \array{ 1 & 0 \\ 0 & 1 } \right] ,\,\; \sigma_1 \;\coloneqq\; \left[ \array{ 0 & 1 \\ 1 & 0 } \right] ,\,\; \sigma_2 \;\coloneqq\; \left[ \array{ 1 & 0 \\ 0 & -1 } \right] ,\,\; \sigma_3 \;\coloneqq\; \left[ \array{ 0 & -i \\ i & 0 } \right] \,.$

In quantum information theory one often considers the higher Pauli group $\mathcal{P}_n$ whose elements are (multiples by $\pm 1$, $\pm i$) of $n$-fold tensor products of the Pauli matrices.

## Properties

### Relation to the quaternion group

The quaternion group $Q$ is the (normal) subgroup of the Pauli group (1) omitting the $\pm 1$-phases on nonidentity matrices:

(2)$Q \;\simeq\; \big\{ \pm \sigma_0, \, \pm i\sigma_1, \, \pm i\sigma_2, \, \pm i\sigma_3, \big\} \;\subset\; GL(2,\mathbb{C}) \,,$

In fact, the Pauli group is a semidirect product group of the quaternion group with the cyclic group of order 2:

$\mathcal{P}_1 \;\simeq\; Q \rtimes \mathbb{Z}/2 \,.$

See also:

Discussion in the context of quantum error correction codes:

• Simeon Ball, Aina Centelles, Felix Huber, p. 8 of: Quantum error-correcting codes and their geometries (arXiv:2007.05992)