Pauli matrices

The Pauli matrices are three small two-by-two arrays of numbers, written σx, σy, and σz, that Wolfgang Pauli introduced to handle electron spin. Because a spin-½ state has just two basis states, up and down, every operation on it can be written as a two-by-two matrix, and these three are the basic building blocks. The spin components along the three spatial axes are each one of these matrices multiplied by half the reduced Planck constant.

Despite their plainness they carry the whole strangeness of spin. They do not commute, meaning the order in which you apply them matters: measuring spin along x and then along z gives different statistics from doing it the other way round. This non-commuting is the precise mathematical statement that the three spin directions are incompatible observables that cannot all have definite values at once.

The Pauli matrices reach far beyond spin. Together with the identity they form a complete toolkit for any two-level quantum system, so they are the everyday language of qubits and quantum computing. Rotations of a spin, the gates of a quantum circuit, and the description of light's polarisation are all written compactly in terms of these same three matrices.