Optimal Bound on Long-Range Distillable Entanglement

We prove an upper bound on long-range distillable entanglement in D spatial dimensions. Namely, it must decay faster than $1/r$, where r is the distance between entangled regions. For states that are asymptotically rotationally invariant, the bound is strengthened to 1/rD. We then find explicit examples of quantum states with decay arbitrarily close to the bound. In one dimension, we construct free fermion Hamiltonians with nearest neighbor couplings that have these states as ground states. Curiously, states in conformal field theory are far from saturation, with distillable entanglement decaying faster than any polynomial.