nLab
Pauli group

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Contents

Definition

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

(1)𝒫 1{±σ 0,±σ 1,±σ 2,±σ 3,±iσ 0,±iσ 1,±iσ 2,±iσ 3,}GL(2,), \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

σ 0[1 0 0 1],σ 1[0 1 1 0],σ 2[1 0 0 1],σ 3[0 i i 0]. \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 𝒫 n\mathcal{P}_n whose elements are (multiples by ±1\pm 1, ±i\pm i) of nn-fold tensor products of the Pauli matrices.

Properties

Relation to the quaternion group

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

(2)Q{±σ 0,±iσ 1,±iσ 2,±iσ 3,}GL(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:

𝒫 1Q/2. \mathcal{P}_1 \;\simeq\; Q \rtimes \mathbb{Z}/2 \,.

References

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)

Last revised on May 5, 2021 at 08:12:21. See the history of this page for a list of all contributions to it.