Let $ A, B$ be Hermitian matrices. Is it true $ $ \det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)?$ $ As usual, $ |X|=(X^*X)^{1/2}$ . Clearly, quantities on both sides are no less than $ \det(A+B)^2$ .Read more