  In this work, we prove a new family of Lieb-Robinson bounds for lattice spin
systems with long-range interactions. Our results apply for arbitrary $k$-body
interactions, so long as they decay with a power-law greater than $kd$, where
$d$ is the dimension of the system. More precisely, we require that the sum of
the norm of terms with diameter greater than or equal to $R$, acting on any one
site, decays as a power-law $1/R^\alpha$, with $\alpha > d$. These new bounds
allow us to prove that, at any fixed time, the spatial decay of quantum
information follows arbitrarily closely to $1/r^{\alpha}$. Moreover, we define
a new light-cone for power-law interacting quantum systems, which captures the
region of the system where changing the Hamiltonian can affect the evolution of
a local operator. In short-range interacting systems, this light-cone agrees
with the conventional definition. However, in long-range interacting systems,
our definition yields a stricter light-cone, which is more relevant in most
physical contexts.
