Two sequentially generated locally purifiable density operators describe the same state for all system sizes exactly when their purification tensors are linked by a matrix product isometry, under step-injective or cyclic conditions.
A proof is provided in appendix A
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Structure of matrix product locally purifiable density operators
Two sequentially generated locally purifiable density operators describe the same state for all system sizes exactly when their purification tensors are linked by a matrix product isometry, under step-injective or cyclic conditions.