2017/07/31 by Gemma De las Cuevas, J. Ignacio Cirac, Norbert Schuch +1 · 1 citation
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1063/1.5000784
published as Journal of Mathematical Physics 58, 121901 (2017) · 12 pages
arxiv created 2017/07/31 · arxiv updated 2018/04/17
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.