Paulimatriisit

Paulimatriisit, known in English as Pauli matrices, are a set of three 2x2 complex Hermitian matrices that encode the spin-1/2 degrees of freedom in quantum mechanics and serve as the generators of the SU(2) Lie group. They are denoted σ_x, σ_y, and σ_z. The explicit form is: σ_x = [[0, 1], [1, 0]], σ_y = [[0, -i], [i, 0]], σ_z = [[1, 0], [0, -1]]. Each matrix squares to the identity, is Hermitian and has trace zero; each has eigenvalues ±1.

The Pauli matrices satisfy the algebra {σ_i, σ_j} = 2δ_ij I and σ_i σ_j = δ_ij I + i ε_ijk σ_k, where δ_ij is the Kronecker delta and ε_ijk is the Levi-Civita symbol. They form a basis for the su(2) Lie algebra and provide representations of spin operators S_i = (ħ/2) σ_i.

In practice, the matrices are used to describe spin measurements along the x, y, and z axes, to construct rotations U(n, θ) = exp(-i θ n · σ / 2), and to represent quantum states on the Bloch sphere as ρ = 1/2(I + r · σ). In quantum information, the Pauli matrices are the three fundamental single-qubit gates, commonly labeled X, Y, and Z.

The concept is named after Wolfgang Pauli, who introduced them in the context of quantum mechanics in the 1920s. They remain a standard tool in physics and quantum computing.