Fundamental theorem of matrix product states (fundamental theorem of MPS)

Two injective uniform MPS tensors generate the same states on every sufficiently long periodic chain exactly when they differ by a virtual similarity transform and an overall phase: $A^i=e^{i\theta }X{B}^i{X}^{-1}$.

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.