Let $A,B \in \mathbb{C}^{n \times n}$ be given $A,B \neq 0$. Then I would like to know what 

$$\inf_{V_1,...,V_n \in \mathbb{C}^{n \times n}} \left\lVert AB - \sum_{k=1}^{n} V_k B V_k^* \right\rVert$$ is.

So I would like to know, if we can say in general for two matrices, how much conjugation is contained in multiplying $A$ and $B$.

It is easy to get an upper bound:
$V_1=A$ and $V_2=...=V_n=0.$ 
This way,  $$\inf_{V_1,...,V_n} \left\lVert AB - \sum_{k=1}^{n} V_k B V_k^* \right\rVert\le \left\lVert AB(1-A^*)\right\rVert. $$


Is there a way to get more elaborate bounds? Has this question been studied somewhere? I am curious. Thanks a lot