using this spin operator and ladder operators. They can be found in . The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. Also useful in the quantum mechanics of multiparticle systems, the general Pauli group is defined to consist of all -fold tensor products of Pauli matrices. Relativistic quantum mechanics In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as It follows from this definition that the matrices have the same algebraic properties as the matrices. However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence needs to be replaced by the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the are also antisymmetric. Hence there are only six independent matrices. The first three are the The remaining three, where the Dirac matrices are defined as The relativistic spin matrices are written in compact form in terms of commutator of gamma matrices as Quantum information In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the "Z–Y decomposition of a single-qubit gate". Choosing a different Cartan pair gives a similar "X–Y ''decomposition of a single-qubit gate". See also Algebra of physical space Spinors in three dimensions Gamma matrices Angular momentum Gell-Mann matrices Poincaré group Generalizations of Pauli matrices Bloch sphere Euler's four-square identity For higher spin generalizations of the Pauli matrices, see Exchange matrix (the first Pauli matrix is an exchange matrix of order two) Remarks Notes References Lie groups Matrices Rotational symmetry Articles containing proofs Mathematical physics