Partial trace inequalities for partial transpose of positive semidefinite block matrices

Li (Algebra 71:2823–2838, 2023) recently obtained several improvements on some partial trace inequalities for positive semidefinite block matrices. In this note, we present analogous partial trace inequalities involving partial transpose of positive semidefinite block matrix. The inequalities we show could be regarded as complements of Li’s results. In addition, some new partial trace inequalities for partial transpose of positive semidefinite block matrix are included.