Matrix product states, geometry, and invariant theory

TL;DR

This paper explores the geometric and invariant-theoretic properties of matrix product states, linking quantum state representations with classical matrix polynomial identities, especially for 2x2 matrices.

Contribution

It establishes a connection between matrix product states and invariant theory, providing a formula for the dimension of their linear span in the 2x2 case.

Findings

Derived a formula for the dimension of the linear span of homogeneous matrix product states for 2x2 matrices.

Linked the study of matrix product states to classical polynomial identities of matrices.

Explored the geometric structure of matrix product states as subvarieties in tensor spaces.

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.