Pauli-matrix representation of the Affleck--Kennedy--Lieb--Tasaki state

In a Cartesian basis of the physical spin-one space, the AKLT MPS matrices are proportional to ${\sigma }_x,{\sigma }_y,{\sigma }_z$. Products of two such matrices span the full virtual matrix algebra, and the two-site parent term projects onto total spin two.

Because ${\sigma }_i{\sigma }_j={\delta }_{ij}I+i{\epsilon }_{ijk}{\sigma }_k$, the two-site image of the Pauli-matrix AKLT tensor is the direct sum of the scalar and antisymmetric subspaces, which have total spin zero and one. Its orthogonal complement is the symmetric traceless total-spin-two subspace.