Pauli matrices

The Pauli matrices are defined as σ₁ = [[0,1],[1,0]], σ₂ = [[0,-i],[i,0]], and σ₃ = [[1,0],[0,-1]], each of which is Hermitian (σ_i^† = σ_i), unitary (σ_i² = I), and traceless (Tr(σ_i) = 0).¹ They satisfy the commutation relation [σ_i, σ_j] = 2iε_ijkσ_k and the anticommutation relation {σ_i, σ_j} = 2δ_ijI, where ε_ijk is the Levi-Civita symbol and δ_ij is the Kronecker delta.³ Any 2×2 Hermitian matrix can be expressed as a linear combination of the Pauli matrices together with the identity matrix, making them a complete basis for the real vector space of 2×2 Hermitian operators.² The matrices are also the generators of the Clifford algebra Cl(3,0) and correspond to the fundamental representation of the Lie algebra su(2).

In quantum mechanics, the Pauli matrices represent the spin-½ operators for an electron, with the eigenvalues ±1 corresponding to spin up and spin down along the respective axes.² The Pauli equation, which describes a non-relativistic spin-½ particle in an electromagnetic field, incorporates the Pauli matrices as the spin coupling term.⁴ The expectation values of the Pauli matrices along a given direction define the Bloch vector, which fully characterizes the state of a two-level quantum system (qubit).¹ The matrices also appear in the Dirac equation through the representation of the Dirac matrices in (3+1)-dimensional spacetime, where they generate the spinor rotations.⁵

Beyond their foundational role in quantum theory, the Pauli matrices are essential in quantum computing as the single-qubit Pauli gates (X, Y, Z).¹ They form a basis for the Lie algebra su(2), and the exponential map exp(iθσ_i/2) generates rotations in SU(2), which double-cover the rotation group SO(3).³ The matrices satisfy the identity (σ·a)(σ·b) = (a·b)I + iσ·(a×b), relating to the Pauli vector σ = (σ₁, σ₂, σ₃). This identity is used in the derivation of the Fermion anticommutation relations and in the algebraic formulation of the Dirac equation. The Pauli matrices also appear in the representation of quaternions: the unit quaternions correspond to linear combinations of the Pauli matrices with real coefficients, mapping the group SU(2) to the unit quaternions.⁵

Though often introduced as a set of three, it is common to include the identity matrix σ₀ = I to form a basis for all 2×2 complex matrices, yielding the four-dimensional real vector space of Hermitian operators.¹ The eigenvalues of any linear combination n·σ, where n is a unit vector, are ±1, a fact that underlies the Bloch sphere representation of qubits.³ The Pauli matrices are the simplest case of the Dirac matrices in 2 dimensions and are directly related to the Rodrigues' rotation formula for rotations in 3D. A less-known application is in the description of neutrino oscillations, where the Pauli matrices parameterize the mixing matrix. The matrices also appear in the theory of electron diffraction and in the topological classification of insulators via the Berry phase.⁵ Wolfgang Pauli's original 1927 paper introduced them in the context of the magnetic electron, and the matrices have since become a staple in both pure and applied mathematics, especially in representation theory and differential geometry.⁴

Hermitian matrix

A complex square matrix equal to its own conjugate transpose.

Unitary matrix

A complex square matrix whose inverse equals its conjugate transpose.

Traceless matrix

A matrix whose trace (sum of diagonal entries) is zero.

Spin (physics)

An intrinsic form of angular momentum carried by elementary particles.

SU(2)

The special unitary group of degree 2, consisting of 2×2 unitary matrices with determinant 1.

Clifford algebra

An algebra generated by a vector space with a quadratic form, used in the construction of spinors.

Pauli vector

The vector of Pauli matrices, σ = (σ₁, σ₂, σ₃).

Qubit

A basic unit of quantum information, represented by a two-level quantum system.