Pauli matrices

The Pauli matrices are three 2×2 complex Hermitian matrices that represent spin components for spin-½ particles and form a basis for traceless Hermitian 2×2 matrices.

Pauli algebra: The multiplication and commutation rules obeyed by the Pauli matrices. Their products combine a scalar identity term with another Pauli matrix, fixing the algebra's structure.
Hermitian matrix: A complex square matrix equal to its conjugate transpose. Each Pauli matrix is Hermitian, so its eigenvalues can represent observable measurement outcomes.
Quantum computing: Computation that uses quantum states and operations to process information. Single-qubit operations are commonly expressed using Pauli matrices and their exponentials.
Identity matrix: A square matrix that leaves every vector unchanged under multiplication. It completes the Pauli matrices to a basis for all 2×2 complex matrices but is not traceless.
Spin operator: An observable representing intrinsic angular momentum in quantum mechanics. Each spin component for a spin-½ particle is one-half of the corresponding Pauli matrix times ħ.
Trace (linear algebra): The sum of the diagonal entries of a square matrix. All three matrices have zero trace, distinguishing them from the identity matrix.
Quantum gate: A unitary operation that transforms a quantum state during a computation. The X, Y, and Z gates are the Pauli matrices used directly as single-qubit gates.
Gell-Mann matrices: Eight traceless Hermitian 3×3 matrices used as generators of SU(3). They generalize the role of Pauli matrices from two-level systems to three-level symmetry.
Bloch sphere: A unit sphere representation of pure states of a two-level quantum system. Expectation values of the three Pauli matrices give a state's coordinates on this sphere.
Eigenvalue: A scalar by which a linear transformation scales one of its eigenvectors. The eigenvalues ±1 yield spin-component outcomes of ±ħ/2 after the physical scaling.

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