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 d, its Gibbs state at inverse temperature β, denoted by ρ =e-β H/ tr(e-β H), is a classical distribution over product states for all β < 1/(cd), where c is a constant. This sudden death of thermal entanglement upends conventional wisdom about the presence of short-range quantum correlations in Gibbs states. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any β < 1/( c d³), we can prepare a state ε-close to ρ in trace distance with a depth-one quantum circuit and poly(n) log(1/ε) classical overhead. A priori the task of preparing a Gibbs state is a natural candidate for achieving super-polynomial quantum speedups, but our results rule out this possibility above a fixed constant temperature.