Border rank is not multiplicative under the tensor product

TL;DR

This paper proves that border rank, a measure of tensor complexity, can be strictly submultiplicative under tensor product, answering an open question and extending understanding of tensor rank behaviors.

Contribution

It demonstrates that border rank can be strictly submultiplicative, providing explicit constructions and linking to previous tensor rank drop results.

Findings

Border rank is strictly submultiplicative under tensor product.

Constructs lines in projective space where border rank drops multiple times.

Implicates strict submultiplicativity for cactus rank and border cactus rank.

Abstract

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.