High-Temperature Gibbs States are Unentangled and Efficiently Preparable

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian H on a graph with degree mathfrakd, its Gibbs state at inverse temperature β, denoted by ρ = e-β H/ operatornametr(e-β H), is a classical distribution over product states for all β < 1/(cmathfrakd), where c is a constant. This proof of sudden death of thermal entanglement resolves the fundamental question of whether many-body systems can exhibit entanglement at high temperature. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any β < 1/( c mathfrakd2), we can prepare a state varepsilon-close to ρ in trace distance with a depth-one quantum circuit and operatornamepoly(n, 1/varepsilon) classical overhead.