Scalet proves a finite entanglement length for one-dimensional spin-chain Gibbs states at any finite temperature, so left and right half-chains become exactly separable once a long enough interval is traced out

The paper proves a system-size-independent finite entanglement length: for a local interaction of range r and strength J on a one-dimensional spin chain, there is a function ℓ(d,J,r) such that for any ordered contiguous intervals A, B, C with |B| ≥ ℓ(d,J,r), the marginal ρAC of the Gibbs state ρABC = exp(−HABC)/ZABC is separable between A and C, with ℓ independent of |A| and |C|. Earlier results on entanglement in one-dimensional thermal states (such as KS22 and Kuw24) established decay of entanglement or mutual information rather than exact vanishing, and the high-temperature result of Bakshi et al. applies only in the trivial high-temperature phase with a separable ball whose size may grow with dimension. This work upgrades exact separability from high temperature to any fixed finite temperature and provides the first entanglement length that does not grow with system size. This is a complete mathematical proof: approximate factorization ‖ρAC − ρA ⊗ ρC‖ ≤ C exp(−α|B|) follows from the correlation-decay result of Bergamaschi–Chen via the equivalences of BCP22; a positive decomposition comes from the exponential lower bound on the smallest eigenvalue of marginals (Lemma 6, ‖ρB^{-1}‖ ≤ C exp(α|B|)); the tail terms ∆k are controlled by the superexponential locality of Araki expansionals (Lemma 4); and the separable ball around the maximally mixed state of Gurvits–Barnum (Lemma 2) closes the argument.

The paper gives an intermediate constant-size result (Proposition 1): there is ℓ1(k) = O(k) such that for |B| ≥ ℓ1(k), e^{H^k_AC/2} ρ^k_AC e^{H^k_AC/2} = γ(k)1 + Γ(k), where γ(k) = exp(−O(k))1 and Γ(k) is separable. This proposition shows that separability needs only two ingredients, uniform faithfulness of marginals (a uniform lower bound on all constant-sized marginals) and decay of correlations; the author notes that one-dimensionality is not essential here, so the argument can be extended wherever uniform faithfulness and correlation decay can be proven. The proof is constructive: the deviation ∆ is bounded as C exp(2Jk − α|B|) via approximate factorization, the marginal lower bound is written as C′ exp(−α′k)1, and choosing γ(k) = C′² exp(−2α′k)/2 makes the first three terms separable by positivity, while the remaining term γ(k)1 + ∆ is separable by Lemma 2 once ‖∆‖/γ(k) ≤ d^{−k}, giving |B| ≥ ℓ1 = O(k).

The paper lifts the finite-interval result to the thermodynamic limit (Corollary 2): for an interaction of range r and strength J there is ℓ ∈ N depending only on r, J, and d such that the KMS state ω of the infinite system, after the partial trace tr_{[−ℓ,0]}, is separable between the left and right half-chain algebras. Because the main theorem holds uniformly in system size, this corollary is direct; the author deliberately uses an explicit construction (defining ω_m via ρ_{[−m,m]} ⊗ 1/d^{2m+1} and taking the weak-* limit) to avoid the possibility that a Hahn–Banach extension introduces extra correlations or entanglement between the left and right subsystems. The proof relies on uniqueness of the KMS state for one-dimensional finite-range interactions (Araki 1975) and on closedness of the set of separable operators in the weak-* topology: each ω_m ∘ tr_{[−ℓ,0]} decomposes by Theorem 2 into a finite convex combination of product states, and product structure is preserved when passing to limits on a dense subset.

The paper names the phenomenon a spatial sudden death of entanglement and distinguishes it from the time-domain sudden death of Yu–Eberly and from the analogous phenomenon in high-temperature Gibbs states: here separability is exact and occurs at sufficiently large spatial distance rather than in time or temperature. The author notes that entanglement between neighboring spins is unavoidable, so separability between arbitrary pairs of spins cannot be proven; this is therefore a new notion of sudden death parameterized by distance, and the author describes it as the first nontrivial example of its kind. This is a conceptual positioning and literature comparison, grounded in the paper's discussion of YE04, YE09, Bak+24, KS22, Kuw24, and BCP26 rather than in independent numerical or experimental evidence.

The result applies to one-dimensional spin chains with finite-range, bounded-strength local interactions at any fixed finite temperature (β fixed), and the entanglement length ℓ depends only on temperature and locality, not on system size; the author notes that ℓ grows with β and that, with the techniques used, the dependence should take the form exp(exp(O(β))). The conclusion holds both for finite intervals and for the KMS state of the infinite system, so it applies directly to the left and right half-chains in the thermodynamic limit. For a reader, this means that beyond a sufficiently large spatial scale the remaining correlations in the thermal state can be treated as purely classical, and that this marginal state can be prepared using only local operations and classical communication. The author also notes that the one-dimensional assumption is not essential for the constant-size proposition (Proposition 1), so the argument can migrate wherever uniform faithfulness and decay of correlations can be established.

Several open directions remain of interest: the author asks whether a superexponential decay of conditional mutual information can be shown building on these techniques, and notes that in the classical setting the stronger Hammersley–Clifford theorem implies exact Markovianity beyond the interaction range; whether the sudden death of entanglement extends to higher-dimensional low-temperature systems or ground states is described as beyond the scope of these techniques, though monogamy of entanglement might offer a hint, and states prepared by constant-depth quantum circuits already exhibit the death of entanglement while maintaining short-range entanglement. In addition, the precise dependence of the entanglement length on β (expected by the author to be exp(exp(O(β)))) is not worked out in detail, so the actual numerical scale remains to be clarified.