High-Temperature Gibbs States Are Unentangled and Efficiently Preparable

We show that thermal states of local Hamiltonians are separable above aconstant temperature. Specifically, for a local Hamiltonian H on a graph withdegree 𝔡, its Gibbs state at inverse temperature β, denotedby ρ =e^-β H/ tr(e^-β H), is a classical distributionover product states for all β < 1/(c𝔡), where c is aconstant. This sudden death of thermal entanglement upends conventional wisdomabout the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution overproduct states. In particular, for any β < 1/( c 𝔡^3), we canprepare a state ϵ-close to ρ in trace distance with a depth-onequantum circuit and poly(n) log(1/ϵ) classical overhead. Apriori the task of preparing a Gibbs state is a natural candidate for achievingsuper-polynomial quantum speedups, but our results rule out this possibilityabove a fixed constant temperature.