Fundamental conjecture of PEPS equivalence via intertwining MPOs

Establish a complete fundamental theorem for projected entangled-pair states (PEPS) that provides necessary and sufficient conditions under which two PEPS tensors A and B generate the same state for all system sizes; specifically, prove that the existence of a matrix product operator (MPO) satisfying the pulling-through relations with A and B (as described by the intertwining conditions) is both necessary and sufficient for such global state equivalence.

Background

For one-dimensional systems, the fundamental theorem of matrix product states (MPS) gives necessary and sufficient conditions—via a gauge transform—for two MPS tensors to represent the same state. In higher dimensions, the analogous result for PEPS is not generally available.

The notes identify a clear sufficient condition for PEPS equivalence: the existence of an MPO that intertwines the local PEPS tensors and can be pulled through the network. They explicitly name the unproven necessity of this condition the "fundamental conjecture of PEPS." Proving this would unify symmetry considerations and equivalence in PEPS, similar to the MPS case.