Uniform matrix product states with a boundary

TL;DR

This paper extends the canonical form of uniform matrix product states (MPS) to include boundary conditions, providing a unified framework that broadens the analytical understanding and classification of MPS beyond periodic boundaries.

Contribution

It introduces a generalized canonical form for uniform MPS with a boundary, connecting boundary states to algebraic regular language states and establishing new algebraic results.

Findings

Unified theoretical foundation for boundary-including MPS

Explicit bounds on algebraic stabilization length

Bridging gap between periodic and arbitrary-boundary MPS

Abstract

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.