Let $Q_n= \begin{pmatrix} a_n & 0 \\ 0 & b_n \end{pmatrix}$, where $a_n\neq b_n$ and $a_n, b_n\in \Bbb C$ satisfying $a_n\rightarrow a,b_n\rightarrow b$.
Can we find a sequence of invertible matrices $\{P_n\}$ such that $B_n:=P_nQ_nP_n^{-1}$ with $tr(B_n^*B_n)-|a_n|^2-|b_n|^2\geq 1$ for all $n$, where $tr(B_n^*B_n)$ denotes the trace of $B_n^*B_n$.