Irreducible forms of matrix product states: Theory and applications

• Published in: Journal of mathematical physics 2017-12, Vol.58 (12), p.1
• Main Authors: De las Cuevas, Gemma, Cirac, J. Ignacio, Schuch, Norbert, Perez-Garcia, David
• Format: Article
• Language: English
• Subjects: Antiferromagnetism; Mathematical analysis; Matrix; Matrix methods; Physics; Symmetry; Tensors; Theorems
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.5000784

The canonical form of Matrix Product States (MPS) and the associated fundamental theorem, which relates different MPS representations of a state, are the theoretical framework underlying many of the analytical results derived through MPS, such as the classification of symmetry-protected phases in one dimension. Yet, the canonical form is only defined for MPS without non-trivial periods and thus cannot fully capture paradigmatic states such as the antiferromagnet. Here, we introduce a new standard form for MPS, the irreducible form, which is defined for arbitrary MPS, including periodic states, and show that any tensor can be transformed into a tensor in irreducible form describing the same MPS. We then prove a fundamental theorem for MPS in irreducible form: If two tensors in irreducible form give rise to the same MPS, then they must be related by a similarity transform, together with a matrix of phases. We provide two applications of this result: an equivalence between the refinement properties of a state and the divisibility properties of its transfer matrix, and a more general characterisation of tensors that give rise to matrix product states with symmetries.