Tensor rank of the direct sum of two copies of $2 \times 2$ matrix multiplication tensor is 14

TL;DR

This paper proves that for certain small three-way tensors, the tensor rank additivity holds, specifically confirming Strassen's conjecture for the case of two $2 imes 2$ matrix multiplication tensors, and discusses the preservation of structure by a substitution method.

Contribution

It demonstrates the validity of Strassen's additivity conjecture for the specific case of two $2 imes 2$ matrix multiplication tensors and analyzes the structure-preserving properties of a substitution method.

Findings

Additivity holds for some small three-way tensors.

Strassen's conjecture confirmed for two $2 imes 2$ matrix multiplication tensors.

Substitution method preserves tensor direct sum structure.

Abstract

The article is concerned with the problem of the additivity of the tensor rank. That is for two independent tensors we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we show that for some small three-way tensors the additivity holds. For instance, we give a proof that another conjecture stated by Strassen (1969) is true. It is the particular case of the general Strassen's additivity conjecture where tensors are a pair of $2 \times 2$ matrix multiplication tensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.