Long-range quantum Gibbs-state separability threshold

The problem asks whether sufficiently high temperature destroys entanglement in long-range quantum spin Gibbs states uniformly in system size, and how the threshold depends on interaction degree and locality. A recent preprint answers this for several structured Hamiltonian families.

Known results

• A 2024 result places high-temperature unentanglement at inverse temperature of order $1/(cd)$, but does not establish the sharp long-range threshold or its locality dependence.

July 2026 threshold theorem

For long-range Pauli Hamiltonians, the preprint gives $\eta_{ m sep}=\Theta(1/(sk))$, with separability guaranteed for $0\leq\beta\leq 1/(72sk)$. Matching entangled examples show this scaling is tight, including commuting systems, for every even $k\geq 2$ and every $s>0$. It also treats $(d,k)$-low-intersection families, though the supplied record omits the full formula.

Current status (as of July 2026): The size-independent, sharp scaling is settled for the stated long-range Pauli families, while the corresponding theorem for unrestricted long-range quantum spin systems remains open or requires a precise interaction class.