Can a sequence of operators with trace-class norm 1 have a trace that converges absolutely to 0?

Question

Does there exist a sequence of (non-normal) trace-class operators $X_n$ such that in some ONB $\{ v_i \}_{i \in \mathbb{N}}$ we have \begin{align*} \sum_{i=1}^\infty \vert \langle v_i, X_n v_i \rangle \vert \to 0 \text{ as } n \to \infty \end{align*} and at the same time $\vert \vert X_n \vert \vert_{tc} = 1 $ for all $n \in \mathbb{N}$?

Answer 1

One can consider $X_{ij}^n = 1_{i = j = n}$.