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Normalize the stated Pauli matrices asThe factor $3^{-1/2}$ changes only the overall normalization at fixed $N$. In the spin-one Cartesian basis, the associated periodic uniform matrix product state isThis is the Pauli-matrix representation of the Affleck--Kennedy--Lieb--Tasaki state.

The Pauli matrix multiplication lawshows that products on two neighboring sites span all of $M_2(\mathbb{C})$, so the tensor is an injective matrix product state after blocking two sites. More geometrically, its two-site image is the scalar plus antisymmetric subspace of $3\otimes 3$, namely the total-spin $J=0$ and $J=1$ sectors. By the Two-site support of the Pauli-matrix Affleck--Kennedy--Lieb--Tasaki tensor, the missing subspace is the five-dimensional symmetric traceless $J=2$ sector.

Let $P_{n,n+1}^{(2)}$ be the orthogonal projection onto that $J=2$ sector. The parent Hamiltonian of a matrix product state isEach term annihilates $|{\Psi }_N⟩$, so this is a frustration-free quantum Hamiltonian and the MPS is a ground state. Writing $x={\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$ and using the eigenvalues $-2,-1,1$ in the three total-spin sectors gives the explicit projectorThis is the Affleck--Kennedy--Lieb--Tasaki parent Hamiltonian. Injectivity implies that its periodic ground state is unique for every sufficiently long chain.

For the injective translationally invariant case, the fundamental theorem of matrix product states states that two tensors $A=\{A^i\}$ and $B=\{B^i\}$ of the same minimal bond dimension generate the same periodic MPS for every sufficiently large length if and only iffor every physical index $i$, with one invertible matrix $X$. Equality as normalized rays permits the phase $e^{i\theta }$; equality as vectors for all lengths restricts the resulting factor $e^{iN\theta }$ accordingly. The converse is immediate from cyclicity of the matrix trace:

For the nontrivial direction, block enough sites that both tensors are injective and definewith ${\Gamma }_B^L$ defined similarly. Injectivity means that ${\Gamma }_A^L$ and ${\Gamma }_B^L$ have trivial kernels. Equality of all sufficiently long periodic states implies equality of the local support spaces $\mathrm{im}{\Gamma }_A^L=\mathrm{im}{\Gamma }_B^L$, so there is an invertible linear map $F$ on the virtual matrix algebra satisfying

Compare two adjacent blocks and contract arbitrary environments on their left and right. Because both block maps are injective, equality of the physical contractions forcesThus $F$ is a unital algebra automorphism of $M_{\chi }(\mathbb{C})$. By every automorphism of a full matrix algebra is inner, $F(X)=X_{0}X{X}_0^{-1}$ for an invertible $X_0$. Applying this relation to a block with one physical site exposed givesThis proves the theorem and identifies the freedom as the gauge equivalence of injective matrix product state tensors. For noninjective tensors, their canonical forms first split into injective blocks; equality then permits a permutation of equivalent blocks together with a similarity transformation and phase on each block.