Upper Bounds on the Strassen CP Rank Gap and Symmetric Rank Gap of Third-Order Tensors

Abstract

For two third-order tensors over $\mathbb{R}$ or $\mathbb{C}$, we define the Strassen CP rank gap as the sum of their CP ranks minus the CP rank of their direct sum. For arbitrary third-order tensors, we derive explicit upper bounds on this gap in terms of CP rank excess and dimension. For symmetric third-order tensors, we similarly define the Strassen symmetric rank gap using symmetric rank and establish explicit upper bounds in terms of symmetric rank excess, CP rank excess, and dimension. These bounds also yield sufficient conditions for the respective gaps to vanish, ensuring additivity of CP rank or symmetric rank under direct sums.