Let $A,B \in \mathbb{C}^{n \times n}$ be given $A,B \neq 0$. Then I would like to know what
$$\inf_{V_1,...,V_n \in \mathbb{C}^{n \times n}} \left\lVert AB - \sum_{k=1}^{n} V_k B V_k^* \right\rVert$$$$\inf_{V_1,...,V_d \in \mathbb{C}^{n \times n}} \left\lVert AB - \sum_{k=1}^{d} V_k B V_k^* \right\rVert$$ is, where $d$ is arbitrary.
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: $V_1=A$ andfor $V_2=...=V_n=0.$$d \ge 1.$ This way, $$\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. $$$$\inf_{V_1,...,V_d} \left\lVert AB - \sum_{k=1}^{d} V_k B V_k^* \right\rVert\le \left\lVert AB(1-A^*)\right\rVert. $$
Is there a way to get more elaborate bounds? Has this question been studied somewhere? What is the dependence on $d$? Does the approximation become better for $d$ large? By linear independence, it seems that making $d$ larger than $n^2$ does no longer improve things, but maybe it is enough to consider much smaller numbers $d$. I am curious. Thanks a lot