I have a very naive question:
Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ (depending only on $p$ ) one might have that $$ (\mbox{Tr } |A-B|^p)^{1/p} \leq K_p n. $$
Obviously, $K_p \leq 2$. But what (or there) is a better upper bound for $K_p?$