Equality condition for a matrix inequality with Kronecker product

Quantum information theory has been greatly developed in the past decades, and many theoretical problems are related to matrix theory. We study the equality condition for a matrix inequality, [Formula: see text] rank([Formula: see text] ⊗[Formula: see text]) [Formula: see text] rank([Formula: see text] ⊗[Formula: see text] where [Formula: see text]’s are linearly independent matrices of the same size, and [Formula: see text]’s are linearly independent matrices of the same size. The inequality used to be a conjecture since 2013 and has recently been proven in the paper [Z. Song, L. Chen, Y. Sun and M. Hu, IEEE Trans. Inform. Theory 69 (2023) 2385]. We study several cases such as that [Formula: see text]’s are column vectors and [Formula: see text]’s are of various sizes. It turns out that some cases never satisfy the equality condition.