One parameter generalization of BW inequality and its application to open quantum dynamics

In this paper, we introduce a one parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a $q$-deformed commutator. For $n\times n$ matrices $A$ and $B$, we consider the inequality $$\mathrm{\Re }⟨[B,A],[B,A{]}_q⟩\le c(q)\mathrm{\parallel }A{\mathrm{\parallel }}^2\mathrm{\parallel }B{\mathrm{\parallel }}^2,$$ where $⟨A,B⟩=\mathrm{t}\mathrm{r}(A^{\ast }B)$ is the Hilbert-Schmidt inner product, $\mathrm{\parallel }A\mathrm{\parallel }$ is the Frobenius norm, $[A,B]=AB-BA$ is the commutator, and $[A,B{]}_q=AB-qBA$ is the $q$-deformed commutator. We prove that when $n=2$, or when $A$ is normal with any size $n$, the optimal bound is given by $$c(q)=\frac{(1+q)+\sqrt{2(1+q^2)}}{2}\mathrm{.}$$ We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for $n$ up to $15$ by numerical optimization. When $q=1$, this inequality is exactly BW inequality. When $q=0$, this inequality leads the sharp bound for the $r$-function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.