High-Temperature Gibbs States are Unentangled and Efficiently Preparable

Abstract

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 ${\hat{I}}^2$, denoted by $\ddot{I}=e^{-{\hat{I}}^{2}H}/\mathrm{tr}(e^{-{\hat{I}}^{2}H})$, is a classical distribution over product states for all ${\hat{I}}^2<1/(c\mathfrak{d})$, 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 ${\hat{I}}^2<1/(c{\mathfrak{d}}^2)$, we can prepare a state $\epsilon$-close to $\ddot{I}$ in trace distance with a depth-one quantum circuit and $\mathrm{poly}(n,1/\epsilon )$ classical overhead.

50 pages; v2 mild quantitative improvements, new exposition