Sharp universal death of entanglement threshold for Pauli Hamiltonians

This paper investigates the phenomenon known as the death of entanglement, where quantum systems become fully separable (unentangled) at sufficiently high temperatures. Specifically, the author seeks to determine the exact universal temperature threshold at which Gibbs states of Pauli Hamiltonians transition from being potentially entangled to being a mixture of product Pauli eigenstates, and whether this regime allows for efficient classical sampling.

The author employs an inclusion-exclusion expansion to decompose the Gibbs operator into a sum of separable components. By expressing the Gibbs state as a weighted sum over connected sets of Hamiltonian terms (polymers), the problem of separability is reduced to proving the positivity of a classical independent set polynomial defined on the term-overlap graph. The author uses tree recursion and deletion induction to establish the positivity of this polynomial, identifying a sharp threshold $z_\Delta$ that depends only on the maximum interaction degree $\Delta$.

The paper proves that for any Pauli Hamiltonian with maximum term-overlap degree $\Delta \ge 2$, the Gibbs state is a mixture of product Pauli eigenstates whenever the inverse temperature $\beta$ satisfies $\beta J \le z_\Delta$. This threshold is shown to be sharp; for any $\beta J > z_\Delta$, there exists a commuting Pauli Hamiltonian with the same degree bound that is entangled. Furthermore, the author provides a classical randomized algorithm that, for any temperature strictly below this threshold, produces samples from a distribution over product Pauli eigenstates that approximates the Gibbs state in trace distance in polynomial time.

Understanding the limits of entanglement in thermal states is crucial for quantum computing and many-body physics. This result provides a definitive boundary for when quantum systems lose their entanglement, bridging the gap between quantum statistical mechanics and classical computational complexity. The ability to efficiently sample these states classically below the threshold has significant implications for the simulation of quantum systems.