High-Temperature Gibbs States are Unentangled and Efficiently Preparable
Ainesh Bakshi, Allen Liu, Ankur Moitra, Ewin Tang
TL;DR
The paper proves that Gibbs states of local Hamiltonians are separable above a constant temperature, showing a sudden death of thermal entanglement at physically reasonable temperatures. It introduces a constructive, pinning-based approach that expresses $e^{-\beta H}$ as a convex combination of separable (often stabilizer) operators, enabling efficient classical sampling and enabling preparation of $\rho$ with a depth-1 quantum circuit for sufficiently high temperature. A key technical ingredient is a low-temperature-like propagator expansion with a cluster-expansion flavor, which bounds the growth of correlations and enables controlled approximation via restricted Gibbs states. The results imply that high-temperature Gibbs sampling admits efficient classical algorithms, providing limited room for quantum advantage in this regime, and connect quantum entanglement properties to classical sampling techniques via a rigorous, programmable framework.
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 $β$, 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.