Irreducible forms of Matrix Product States: Theory and Applications

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.

12 pages

References in corpus (4)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• String order and symmetries in quantum spin lattices
• Entanglement, fractional magnetization and long-range interactions

Cited by in corpus (18)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Rapid thermalization of spin chain commuting Hamiltonians
• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• Normal projected entangled pair states generating the same state
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• The minimal canonical form of a tensor network
• Determining non-Abelian topological order from infinite projected entangled pair states
• Continuum limits of Matrix Product States
• Mixed states in one spatial dimension: decompositions and correspondence with nonnegative matrices
• Pivoting through the chiral-clock family
• Quantum Tensor Networks, Stochastic Processes, and Weighted Automata
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Generalised ansatz for continuous Matrix Product States
• Model Checking Applied to Quantum Physics
• Variational methods for characterizing matrix product operator symmetries
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• The product structure of MPS-under-permutations