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$\mathfrak{g}$, its Gibbs state at inverse temperature$\beta$, denoted by$\rho=e^{-\beta H}/\text{tr}(e^{-\beta H})$, is a classical distribution over product states for all$\beta < 1/ (c \mathfrak{{g}})$, 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$\beta < 1/(c\mathfrak{g}^{3})$, we can prepare a state$\varepsilon$-close to$\rho$in trace distance with a depth-one quantum circuit and$\text{poly}(n)\log(1/\varepsilon)$classical overhead.11In independent and concurrent work, Rouzé, França, and Alhambra [37] obtain an efficient quantum algorithm for preparing high-temperature Gibbs states via a dissipative evolution.