Weak Log-majorization of Unital Trace-preserving Completely Positive Maps

Let Phi : M-n -> M-n be a unital trace preserving completely positive map and A is an element of M-n be a positive definite matrix. Weak log-majorization and weak majorization between Phi(A) and A are studied. Determinantal inequalities between Phi(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer's inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.