Uniform matrix product states from an algebraic geometer's point of view

TL;DR

This paper uses algebraic geometry to analyze the structure and properties of uniform matrix product states, addressing questions about their topology, defining equations, and identifiability in quantum physics.

Contribution

It introduces algebraic geometric methods to study uMPS, providing new insights into their mathematical structure and resolving existing conjectures.

Findings

Characterized the topology of the uMPS tensor space

Derived defining equations for uMPS

Established results on the identifiability of uMPS

Abstract

We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch.