We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree $Δ\ge2$. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most $Δ$ 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 $β>z_Δ$, a finite commuting Hamiltonian with maximum overlap degree at most $Δ$ has an entangled Gibbs state. At any fixed $β<z_Δ$ 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.
We determine the exact universal high-temperature separability threshold for Pauli Hamiltonians of bounded degree Δ≥2. If every coefficient in the Hamiltonian has magnitude at most one and each term has overlapping support with at most Δ 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 β>zΔ, a finite commuting Hamiltonian with maximum overlap degree at most Δ has an entangled Gibbs state. At any fixed β<zΔ 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.
Above some temperature every thermal state of a Hamiltonian family is separable, and above another every thermal state is a mixture of stabilizer states, which carry no magic. Putterman, Zlokapa and Cotler conjectured that the second temperature is the lower one, so that heating removes magic from a whole family before entanglement. We prove this ordering for Pauli Hamiltonians of bounded interaction degree or bounded neighbourhood strength, the two families whose separability threshold is known exactly. A Clifford reduction shows that a Gibbs state is a stabilizer mixture whenever that of a canonical all-anticommuting Hamiltonian on the anticommutation graph of its terms is, with equivalence for independent terms, so commuting overlaps drop out of the guarantee. A cluster expansion exploiting anticommutation then shows that every Gibbs state is a stabilizer mixture for βΛ ≤ 0.2183, strictly beyond the sharp separability constant 1/(2e) ≈ 0.1839, and an exactly solvable graph caps the constant at 0.6232. A single Hamiltonian can still order its own transitions the other way. Because the guarantee counts only anticommuting neighbours, it certifies magic-free Gibbs states of the transverse-field Ising, Heisenberg, Kitaev and perturbed toric-code models at temperatures 1.1 to 5.6 times lower than those at which separability is guaranteed.
At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. We give polynomial-time classical algorithms for estimating thermal expectations up to the phase transition (despite both entanglement and magic), and as a corollary resolve a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).
We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local HamiltonianHon a graph with degreeg, its Gibbs state at inverse temperatureβ, denoted byρ=e−βH/tr(e−βH), is a classical distribution over product states for allβ<1/(cg), wherecis 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 anyβ<1/(cg3), we can prepare a stateε-close toρin trace distance with a depth-one quantum circuit andpoly(n)log(1/ε)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.
We prove that the Gibbs state of any spin chain can be exactly decomposed into a mixture of product states over contiguous blocks of qubits with bounded length, at any constant temperature. As a consequence, any such Gibbs state can be prepared by local unitary circuits of constant depth, with efficient classical preprocessing. We place strong limits on the spatial structure of thermal entanglement: both the entanglement depth and entanglement width (measures of multiparticle entanglement) are bounded by constants, while the localizable entanglement (the bipartite entanglement that can be generated with the aid of local measurements and classical communication) vanishes exactly beyond a constant threshold distance. Our result also has implications for quantum thermalization: for any translation-invariant spin-chain Hamiltonian with nondegenerate spectral gaps, typical low-complexity pure states drawn from a maximally entropic ensemble at any constant effective temperature become locally indistinguishable from the Gibbs state upon unitary evolution. Additionally, we obtain an analogous decomposition for fermionic systems: the Gibbs state of any 1D local fermionic Hamiltonian is exactly a mixture of constant-depth local fermionic circuits acting on Gaussian states.
Entanglement is the hallmark of quantum physics, yet its characterization in interacting many-body systems at thermal equilibrium remains one of the most important challenges in quantum statistical physics. We prove that the Gibbs state of any quantum spin chain can be exactly decomposed into a mixture of matrix product states with a bond dimension that is independent of the system size, at any finite temperature. As a consequence, the Schmidt number, arguably the most stringent measure of bipartite entanglement, is strictly finite for thermal states, even in the thermodynamic limit. Our decomposition is explicit and is accompanied by an efficient classical algorithm to sample the resulting matrix product states.
This preprint studies learning an unknown bounded-degree n-qubit Hamiltonian in a known Pauli basis from independent copies of a single-temperature Gibbs state. For fixed locality, interaction degree and fixed positive rational inverse temperature and coefficient bound, it gives a polynomial-resource algorithm using product single-qubit Pauli measurements, without geometry, a gap, correlation decay or a minimum nonzero coefficient. The output has rational coefficients, support inside the true support and total coefficient error at most the requested tolerance with the stated success probability. The temperature-dependent exponents are extremely large; the result is asymptotic. The refinement certificate is formalized in Lean conditionally on explicit analytic inputs. Initialization, iteration, solver, sampling and full resource bounds remain paper proofs. A companion source note preserves the separate hidden-Majorana discussion and its open calibration problem; it is not a proof of that broader learner. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.