and column of the -th Pauli matrix is In this notation, the completeness relation for the Pauli matrices can be written As noted above, it is common to denote the 2 × 2 unit matrix by so The completeness relation can alternatively be expressed as The fact that any Hermitian complex 2 × 2 matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of as above, and then imposing the positive-semidefinite and trace conditions. For a pure state, in polar coordinates, the idempotent density matrix acts on the state eigenvector with eigenvalue +1, hence it acts like a projection operator. Relation with the permutation operator Let be the transposition (also known as a permutation) between two spins and living in the tensor product space This operator can also be written more explicitly as Dirac's spin exchange operator, Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates. SU(2) The group SU(2) is the Lie group of unitary matrices with unit determinant; its Lie algebra is the set of all anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra is the 3-dimensional real algebra spanned by the set . In compact notation, As a result, each can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of , as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is so that As SU(2) is a compact group, its Cartan decomposition is trivial. SO(3) The Lie algebra is isomorphic to the Lie algebra , which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that the are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though and are isomorphic as Lie algebras, and are not isomorphic as Lie groups. is actually