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The virtual similarity transformation $A^i\mapsto X{A}^i{X}^{-1}$ leaves every periodic MPS amplitude invariant by cyclicity of the matrix trace. The fundamental theorem of matrix product states says that, up to an overall phase, this is the only ambiguity for injective uniform tensors.

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.

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.