A note on the gap between rank and border rank

TL;DR

This paper investigates the tensor rank and border rank of specific algebraic tensors, providing counterexamples to existing conjectures and establishing new bounds, including exact rank calculations for certain quantum states.

Contribution

It introduces a sequence of tensors with a large gap between rank and border rank, countering Rhodes' conjecture, and determines the tensor rank of the tensor cube of the three-party W-state.

Findings

Found a sequence of tensors with a large gap between rank and border rank.

Provided a new lower bound on tensor rank of tensor powers of the W-state tensor.

Exactly determined the tensor rank of the tensor cube of the three-party W-state.

Abstract

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor. In addition, we exactly determine the tensor rank of the tensor cube of the three-party W-state tensor, thus answering a question of Chen et al.