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 $\mathfrak{d}$, its Gibbs state at inverse temperature $β$, denoted by $ρ= e^{-βH}/ \operatorname{tr}(e^{-βH})$, is a classical distribution over product states for all $β< 1/(c\mathfrak{d})$, 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 $β< 1/( c \mathfrak{d}^2)$, we can prepare a state $\varepsilon$-close to $ρ$ in trace distance with a depth-one quantum circuit and $\operatorname{poly}(n, 1/\varepsilon)$ classical overhead.