Improvements on some partial trace inequalities for positive semidefinite block matrices

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove celebrated result Choi [Linear Algebra Appl. 516 (2017)]. Our also allows us generalization another Multilinear 66 (2018)]. Furthermore, shall give an improvement on recent Li, Liu and Huang [Operators Matrices 15 (2021)]. In addition, include with some majorization two by matrices, provide related the unitarily invariant norms as well singular values, which can be viewed slight extensions results Lin 459 (2014)] [Electronic J. Linear 31 (2016)].