On the construction of a counterexample to Strassen's rank additivity conjecture

TL;DR

This paper revisits Shitov's counterexample to Strassen's tensor rank additivity conjecture, providing a detailed explanation and an alternative proof to the known disproof of the conjecture.

Contribution

It offers a comprehensive overview of the Strassen problem and presents an alternative proof of Shitov's counterexample to the rank additivity conjecture.

Findings

Confirmed the existence of a counterexample to the conjecture

Provided a detailed explanation of Shitov's construction

Presented an alternative proof of the counterexample

Abstract

The rank additivity conjecture, first formulated by Volker Strassen in 1973, states that the rank of the direct sum of two independent tensors is equal to the sum of their individual ranks. In the last decades, this conjecture has been a central topic in tensor rank theory and its implications for computational complexity. In 2019, Yaroslav Shitov disproved this conjecture in its general form by showing the existence of a counter-example using a dimension counting argument. In this paper, we provide an overview of the Strassen problem and Shitov's work and revisit his counterexample with a detailed explanation, offering an alternative proof.