Canonical forms are central to the analytical understanding of tensor network states, underpinning key results such as the complete classification of one-dimensional symmetry-protected topological phases within the matrix product state (MPS) framework. Yet, the established theory applies only to uniform MPS with periodic boundary conditions, leaving many physically relevant states beyond its reach. Here we introduce a generalized canonical form for uniform MPS with a boundary matrix, thus extending the analytical MPS framework to a more general setting of wider physical significance. This canonical form reveals that any such MPS can be represented as a block-invertible matrix product operator acting on a structured class of algebraic regular language states that capture its essential long-range and scale-invariant features. Our construction builds on new algebraic results of independent interest that characterize the span and algebra generated by non-semisimple sets of matrices, including a generalized quantum Wielandt's inequality that gives an explicit upper bound on the blocking length at which the fixed-length span stabilizes to an algebra. Together, these results establish a unified theoretical foundation for uniform MPS with boundaries, bridging the gap between periodic and arbitrary-boundary settings, and providing the basis for extending key analytical and classification results of matrix product states to a much broader class of states and operators.
Canonical forms are central to the analytical understanding of tensor network states, underpinning key results such as the complete classification of one-dimensional symmetry-protected topological phases within the matrix product state (MPS) framework. Yet, the established theory applies only to uniform MPS with periodic boundary conditions, leaving many physically relevant states beyond its reach. Here we introduce a generalized canonical form for uniform MPS with a boundary matrix, thus extending the analytical MPS framework to a more general setting of wider physical significance. This canonical form reveals that any such MPS can be represented as a block-invertible matrix product operator acting on a structured class of algebraic regular language states that capture its essential long-range and scale-invariant features. Our construction builds on new algebraic results of independent interest that characterize the span and algebra generated by non-semisimple sets of matrices, including a generalized quantum Wielandt's inequality that gives an explicit upper bound on the blocking length at which the fixed-length span stabilizes to an algebra. Together, these results establish a unified theoretical foundation for uniform MPS with boundaries, bridging the gap between periodic and arbitrary-boundary settings, and providing the basis for extending key analytical and classification results of matrix product states to a much broader class of states and operators.
Canonical forms are central to the analytical understanding of tensor network states, underpinning key results such as the complete classification of one-dimensional symmetry-protected topological phases within the matrix product state (MPS) framework. Yet, the established theory applies only to uniform MPS with periodic boundary conditions, leaving many physically relevant states beyond its reach. Here we introduce a generalized canonical form for uniform MPS with a boundary matrix, thus extending the analytical MPS framework to a more general setting of wider physical significance. This canonical form reveals that any such MPS can be represented as a block-invertible matrix product operator acting on a structured class of algebraic regular language states that capture its essential long-range and scale-invariant features. Our construction builds on new algebraic results of independent interest that characterize the span and algebra generated by non-semisimple sets of matrices, including a generalized quantum Wielandt's inequality that gives an explicit upper bound on the blocking length at which the fixed-length span stabilizes to an algebra. Together, these results establish a unified theoretical foundation for uniform MPS with boundaries, bridging the gap between periodic and arbitrary-boundary settings, and providing the basis for extending key analytical and classification results of matrix product states to a much broader class of states and operators.
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.
We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.
Tensor network methods provide powerful analytical and numerical tools for characterizing quantum phases of matter. While the mathematical structure of matrix product states (MPS) is well understood through the MPS fundamental theorem, an analogous understanding for mixed-state tensor networks remains largely absent: if two purification tensors generate the same density matrix, how are they related? In this work, we initiate the study of a fundamental theorem for matrix product locally purifiable density operators (LPDOs) and focus on sequentially generated LPDOs (sLPDOs), a broad subclass admitting an interpretation in terms of successive applications of quantum channels on an initial state. We prove that, under suitable invertibility or cyclic conditions, two sLPDO representations generate the same density matrix for arbitrary system sizes if and only if they are related by a matrix product isometry acting on the purification bonds. Beyond the sLPDO setting, we provide a counterexample that suggests an obstruction to a general fundamental theorem for LPDOs with periodic boundary conditions. Finally, we discuss implications for mixed-state symmetry-protected topological phases, including the possibility of nontrivial phases protected only by weak symmetry conditions.
Abstract One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of G G - and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.
We study the conditions under which Matrix Product States (MPS) or Matrix Product Operators are exact eigenvectors of an extensive local operator, such as a Hamiltonian. By suitably choosing the local operator, this covers a wide range of settings: Exact eigenstates of Hamiltonians, including scar states, exact MPS trajectories for driven quantum systems, steady states of local Lindbladians, generalized symmetries of either Hamiltonians or density matrices, and many more. Our key result is that a local, fixed-size equation---namely, how a single term in the operator acts on a block of tensors---provides a necessary and sufficient condition for exact solutions. This local characterization allows us to obtain the full space of MPS solutions to all of the aforementioned problems; in particular, we exemplify the power of our results by recovering the quantum group symmetries of the XXZ model. We also discuss applications to numerical algorithms with MPS and the generalization of our results to 2D, i.e., projected entangled pair states (PEPS).