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.
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≤zΔ:=arctanh[max0≤x≤1x(1+x1−x)Δ−1]. 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.
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).
Magic, a quantum resource distinct from entanglement, measures departure from states stabilized by products of Pauli operators. Such stabilizer states can be maximally entangled yet admit efficient classical description: quantum advantage requires magic. We develop a powerful framework to evaluate the degree of magic in generic many-body states. We use it to study the fate of magic under thermal and dynamical evolution. In the transverse field Ising model, we find a thermal sudden death of magic that saturates with system size, and occurs after the death of entanglement. We reach similar conclusions in generic spin chains, and in the Kitaev quantum spin liquid on the honeycomb lattice, where the threshold shows an intricate dependence on subregion shape. We then study how magic equilibrates after a quantum quench. In quenches of generic spin chains, the magic of a subregion typically dies before entanglement, contrary to thermal trajectories. An explanation is given in terms of the geometry of non-magical states. As a third type of trajectory, we examine the fate of magic with separation with an appropriate notion of non-local magic and establish a sudden death. Finally, we show how to experimentally witness a volume law of frame-robust magic in quantum materials by using neutron scattering.
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.
Local Markovian noise cannot bring entanglement back, but it can bring magic back. Unlike separability, stabilizer membership is not preserved by local channels, allowing dissipation to push states out of the stabilizer polytope as well as in. Under local amplitude damping, the
n-qubit GHZ family
α∣0n⟩+β∣1n⟩ (
0<α<β) loses its magic at a lower damping strength
γ− and regains it at a higher one
γ+, while entanglement is irreversibly lost at
γe. This magic--entanglement complementarity,
γe+γ+=1 for every
n, reflects a system--environment duality of amplitude damping. Within real phase-covariant Markovian semigroups the phenomenon is mapped out in full: zero-temperature rebirth occurs if and only if
T2>T1, unital dynamics produce no rebirth, and sufficiently weak thermal excitation confines rebirth to a finite magic island ending in a second sudden death. For small
α, the reborn magic resides in a fully separable state with all proper marginals stabilizer, yet parity-syndrome extraction concentrates it onto a single qubit for magic-state distillation, without loss of expected robustness and with optimal per-register yield
α2/2. Local dissipation further divides pure stabilizer states into magic-generators and magic-insulators: at two qubits, the Bell state
∣Φ+⟩ generates magic immediately, while its Bell-state partner
∣Ψ+⟩ remains stabilizer. Together, magic and entanglement reveal a symmetry invisible to either alone.
Measures of non-stabilizerness (``magic'') of a quantum evolution quantify how hard it is to simulate on a classical computer. A natural question is how to extract information about magic from physical correlators, for example at finite temperature or in the ground state. In this paper we show how to measure magic using out-of-time-order correlators evaluated in an arbitrary state. For every full-rank state, and in particular for every Gibbs state, the resulting quantity vanishes if and only if the evolution is Clifford. We establish its basic properties, relate it to the stabilizer entropy of the Choi state, and prove that, even with a single fixed probe operator, it gives a lower bound on the number of
T gates needed to implement the dynamics in any circuit architecture.