I am asking a question related to Lieb-Robinson bound and nonlocality.

As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. $H=\sum_Z H_Z$ where $H_Z$ (with finite dimension) is supported on a set $Z$ and $||H_Z||$ decays rapidly (at least exponential) with the diameter of the set $Z$. Let $A_X$ and $B_Y$ be local operators supported on set $X$ and $Y$ with $dist(X,Y)>0$ (note $dist(X,Y)=min_{i\in X,j\in Y}(i,j)$), the one can show that there is a upper bound

$||[A_X(t),B_Y]||\le c\, e^{-a\,dist(X,Y)}\big[e^{2s|t|-1}\big]$ for some positive constant $c$, $a$, and $s$.

Then, I have two questions: 1) Suppose there is a nonlocal term in the Hamiltonian, i.e. some term $||H_X||$ decays slowly (at most power-lay) with the diameter of the set $Z$, can we show the following statement:

There exist local operators $A_X$ and $B_Y$ with $[A_X,B_Y]=0$ and $dist(X,Y)>0$, and a positive constant $t_0$ and $s>0$, such that for any $t>t_0$ we have a lower bound:

$||[A_X(t),B_Y]||\ge s$. (Answered by Norbert Schuch)

2) Suppose the Lieb Robinson bound is satisfied and one can also define the Lieb Robinson velocity for a Hamiltonian system, can we prove the Hamiltonian is local?