Irreducible forms of Matrix Product States: Theory and Applications

TL;DR

This paper introduces a new irreducible form for Matrix Product States (MPS) that applies to all MPS, including periodic states, and establishes a fundamental theorem relating tensors in this form, enhancing the theoretical framework of MPS.

Contribution

The authors develop a universal irreducible form for MPS applicable to all states and prove a fundamental theorem for these forms, extending the analytical tools for MPS analysis.

Findings

Irreducible form applies to all MPS, including periodic states.

Fundamental theorem relates tensors in irreducible form via similarity transforms and phases.

Applications include characterizing symmetry properties and transfer matrix divisibility.

Abstract

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.