Title: An upper bound on transport,
Abstract: The linear growth of operators in local quantum systems leads to an effective
lightcone even if the system is non-relativistic. We show that consistency of
diffusive transport with this lightcone places an upper bound on the
diffusivity: $D \lesssim v^2 \tau_\text{eq}$. The operator growth velocity $v$
defines the lightcone and $\tau_\text{eq}$ is the local equilibration
timescale, beyond which the dynamics of conserved densities is diffusive. We
verify that the bound is obeyed in various weakly and strongly interacting
theories. In holographic models this bound establishes a relation between the
hydrodynamic and leading non-hydrodynamic quasinormal modes of planar black
holes. Our bound relates transport data --- including the electrical
resistivity and the shear viscosity --- to the local equilibration time, even
in the absence of a quasiparticle description. In this way, the bound sheds
light on the observed $T$-linear resistivity of many unconventional metals, the
shear viscosity of the quark-gluon plasma and the spin transport of unitary
fermions.