Relation between $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(A)\operatorname{Tr}(B)$

Question

I assume this is a rather elementary question but does it exist some kind of inequality between $\operatorname{Tr}(AB)$ and $\operatorname{Tr}(A)\operatorname{Tr}(B)$?

A priori we do not assume special cases for $A,B$ (such as positivity).

Answer 1

Without further assumptions on $\mathbf A$ and $\mathbf B$ the answer is not really. You can't even say much about $\left|\operatorname{tr}(\mathbf{AB})\right|$ vs $\left|\operatorname{tr}(\mathbf{A}) \operatorname{tr}(\mathbf{B})\right|$, since either one of those terms can be zero independently of the other.

As an example, let $\mathbf{A} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Then $\operatorname{tr}(\mathbf{A}) \cdot \operatorname{tr}(\mathbf{A}) = 0$, but $\operatorname{tr}(\mathbf{AA}) = 2$.

On the other hand, let $\mathbf{B} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $\mathbf{C} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$. Then $\operatorname{tr}(\mathbf{B})\cdot \operatorname{tr}(\mathbf{C}) = 1$, while $\operatorname{tr}(\mathbf{BC}) = \operatorname{tr}\mathbf{0} = 0$.