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A uniform matrix product state is specified by matrices $A^i\in M_D$ and, on a periodic chain, has amplitudesIt is injective when, after some blocking length $\ell$, the products $A^{i_1}\cdots A^{i_{\ell }}$ span $M_D$.

The fundamental theorem of matrix product states says that two injective tensors generating the same states for all sufficiently large $N$ satisfyfor one invertible matrix $X$; conversely this relation plainly gives the same periodic states up to the overall phase $e^{iN\theta }$.

For the proof, block enough sites that both tensors are injective. Injectivity gives left inverses from physical blocks to arbitrary virtual matrices. Equality of the states then implies that replacing one blocked tensor inside any sufficiently long network defines an invertible linear map on its two virtual boundary indices. Applying the replacement at two adjacent blocks in either order shows that this boundary map preserves multiplication: $\Phi (MN)=\Phi (M)\Phi (N)$. Every automorphism of the full matrix algebra is inner, so $\Phi (M)=XM{X}^{-1}$. Undoing the blocking gives $A^i=e^{i\theta }X{B}^i{X}^{-1}$, with only an $N$th-root phase left by periodic closure. Equality for consecutive sufficiently large lengths makes that phase independent of $N$ and completes the result.

Apply the theorem to a symmetric injective MPS. The physically transformed tensor $A_g^i={\sum }_{j}U(g{)}_{ij}{A}^j$ generates the same state, so the theorem forcesApplying $g$ and then $h$ shows that $V(g)V(h)$ and $V(gh)$ implement the same gauge transformation. Injectivity makes that gauge unique up to a scalar, hence $V(g)V(h)=\omega (g,h)V(gh)$. Associativity gives the two-cocycle equation for $\omega$.

A continuous symmetry-preserving gapped path changes the tensor and $V(g)$ continuously but cannot change the discrete cohomology class $[\omega ]$ without losing injectivity, breaking the symmetry, or closing the gap. Thus the fundamental theorem of matrix product states turns the virtual projective representation into the invariant classifying one-dimensional symmetry-protected topological phases. With broken symmetry, one first records the permuted ground-state sectors and then applies the same argument to their unbroken subgroup.