  Let an infinite, homogeneous, many-body quantum system be unitarily evolved
for a long time from a state where two halves are independently thermalized.
One says that a non-equilibrium steady state emerges if there are nonzero
steady currents in the central region. In particular, their presence is a
signature of ballistic transport. We analyze the consequences of the current
observable being a conserved density; near equilibrium this is known to give
rise to linear wave propagation and a nonzero Drude peak. Using the
Lieb-Robinson bound, we derive, under a certain regularity condition, a lower
bound for the non-equilibrium steady-state current determined by equilibrium
averages. This shows and quantifies the presence of ballistic transport far
from equilibrium. The inequality suggests the definition of "nonlinear sound
velocities", which specialize to the sound velocity near equilibrium in
non-integrable models, and "generalized sound velocities", which encode
generalized Gibbs thermalization in integrable models. These are bounded by the
Lieb-Robinson velocity. The inequality also gives rise to a bound on the energy
current noise in the case of pure energy transport. We show that the inequality
is satisfied in many models where exact results are available, and that it is
saturated at one-dimensional criticality.
