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arXiv:2212.13806

Matrix product states, geometry, and invariant theory

Tim Seynnaeve

Published Dec 28, 2022• 0 citations
arXivHTMLPDF

Abstract

Matrix product states play an important role in quantum information theory to represent states of many-body systems. They can be seen as low-dimensional subvarieties of a high-dimensional tensor space. In these notes, we consider two variants: homogeneous matrix product states and uniform matrix product states. Studying the linear spans of these varieties leads to a natural connection with invariant theory of matrices. For homogeneous matrix product states, a classical result on polynomial identities of matrices leads to a formula for the dimension of the linear span, in the case of 2x2 matrices. These notes are based partially on a talk given by the author at the University of Warsaw during the thematic semester "AGATES: Algebraic Geometry with Applications to TEnsors and Secants", and partially on further research done during the semester. This is still a preliminary version; an updated version will be uploaded over the course of 2023.

DOI
10.48550/arxiv.2212.13806

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Matrix product states play an important role in quantum information theory to represent states of many-body systems. They can be seen as low-dimensional subvarieties of a high-dimensional tensor space. In these notes, we consider two variants: homogeneous matrix product states and uniform matrix product states. Studying the linear spans of these varieties leads to a natural connection with invariant theory of matrices. For homogeneous matrix product states, a classical result on polynomial identities of matrices leads to a formula for the dimension of the linear span, in the case of 2x2 matrices. These notes are based partially on a talk given by the author at the University of Warsaw during the thematic semester "AGATES: Algebraic Geometry with Applications to TEnsors and Secants", and partially on further research done during the semester. This is still a preliminary version; an updated version will be uploaded over the course of 2023.
We quantify the representational power of matrix product states (MPS) for entangled qubit systems by giving polynomial expressions in a pure quantum state's amplitudes which hold if and only if the state is a translation invariant matrix product state or a limit of such states.For systems with few qubits, we give these equations explicitly, considering both periodic and open boundary conditions.Using the classical theory of trace varieties and trace algebras, we explain the relationship between MPS and hidden Markov models and exploit this relationship to derive useful parameterizations of MPS.We make four conjectures on the identifiability of MPS parameters.
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.
Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state (MPS) factorization) when studying polynomial invariants. As our main example, we consider invariants of MPSs. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Rényi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as MPSs, contain excellent tools that can be applied in the study of invariants.
Transfer matrices and matrix product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.