Sharp universal death of entanglement threshold for Pauli Hamiltonians

Abstract

We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree $\Delta \ge 2$. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most $\Delta$ other terms, the Gibbs state is a mixture of product Pauli eigenstates whenever \[ β\le z_Δ:= \operatorname{arctanh}\left[\max_{0\le x \le 1}x\left(\frac{1-x}{1+x}\right)^{Δ-1} \right]. \] For every $\beta >z_{\Delta }$, a finite commuting Hamiltonian with maximum overlap degree at most $\Delta$ has an entangled Gibbs state. At any fixed $\beta <z_{\Delta }$ strictly below the threshold, a classical polynomial-time algorithm produces samples from a distribution over product Pauli eigenstates approximating the Gibbs state in trace distance.