On the geometry of Tensor Network States of $2 \times N$ Grids

TL;DR

This paper explores the geometric properties of tensor network states on 2 by N grids, addressing open questions, studying their algebraic closure, and providing bounds on tensor ranks.

Contribution

It offers new insights into the geometry of tensor network states on 2 by N grids, partially answers existing questions, and establishes bounds on tensor ranks.

Findings

Partial answer to Verstraete and Rizzi's question for 2 x N grids

Analysis of the Zariski closure of tensor network states on a torus

Explicit tensors with bounds on the border rank of generic tensors

Abstract

We discuss the geometry of a class of tensor network states, called projected entangled pair states in the Physics literature. We provide initial results towards a question of Verstraete and Rizzi regarding the tensor network state of an $M \times N$ grid; we partially answer the question for a $2 \times N$ grid. We also study the $2 \times N$ grid sitting on a torus and provide initial results towards understanding the Zariski closure of the set of tensor network states associated to this graph. Finally, we give explicit tensors that provide optimal (using currently available methods) bounds on the border rank of a generic tensor in the tensor network state of the $2 \times N$ grid.