Narrow Results

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, $K\cdot \mathrm{\text{rank}}({\sum }_{i=1}^K{R}_i\otimes S_i)\ge \mathrm{\text{rank}}({\sum }_{i=1}^K{R}_i\otimes S_i^T),$ where $R_i$’s are linearly independent matrices of the same size, and $S_i$’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. Theory69 (2023) 2385]. We study several cases such as that $R_i$’s are column vectors and $S_i$’s are of various sizes. It turns out that some cases never satisfy the equality condition.