The Pauli Matrices
The Pauli matrices X, Y, Z form a basis for single-qubit operators and generate rotations on the Bloch sphere.
Three fundamental operators
The Pauli matrices are three 2x2 Hermitian, unitary matrices that appear throughout single-qubit quantum mechanics. With the identity I they span the space of all 2x2 matrices, so any single-qubit operator is a combination of them.
Pauli Y equals [[0, -i],[i, 0]]. X swaps |0> and |1>; Z leaves |0> alone and flips the sign of |1>; Y does both a bit-flip and a phase-flip together.
Algebraic properties
- Each squares to the identity: X^2 = Y^2 = Z^2 = I
- They anticommute: XY = -YX, and cyclically XY = iZ
- They are Hermitian (observables) and unitary (gates) at once
As observables
Because they are Hermitian with eigenvalues +1 and -1, the Paulis are observables. Measuring Z is a computational-basis measurement; measuring X is a measurement in the {|+>,|->} basis. Expectation values of X, Y, Z are exactly the three components of the Bloch vector.
As rotation generators
Exponentiating a Pauli gives a rotation: exp(-i theta X/2) rotates the Bloch sphere by angle theta about the x axis, and similarly for Y and Z. Every single-qubit unitary is such a rotation, so the Paulis generate the whole single-qubit gate set. They also label the elementary errors — bit-flip (X), phase-flip (Z), and both (Y) — in error correction.