Lower bounds on high-temperature diffusion constants from quadratically extensive almost conserved operators

We prove a general theorem which provides a strict lower bound on high-temperature Green-Kubo diffusion constants in locally interacting quantum lattice systems, under the assumption of existence of a quadratically extensive almost conserved quantity - an operator whose commutator with the lattice Hamiltonian is localized on the boundary sites only. We explicitly demonstrate and compute such a bound in two important models in one dimension, namely in the (isotropic) Heisenberg spin 1/2 chain and in the fermionic Hubbard chain.