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arXiv:2602.13386

Entanglement in quantum spin chains is strictly finite at any temperature

Ainesh Bakshi, Soonwon Choi, Saúl Pilatowsky-Cameo

quant-phcond-mat.stat-mechmath-phSubmitted Feb 13, 2026• Announced Feb 17, 2026• 0 citations
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Abstract

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.

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32 pages, 1 figure

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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.
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.
Efficient methods to access the entanglement of a quantum many-body state, where the complexity generally scales exponentially with the system size N, have long been a concern. Here we propose the Schmidt tensor network state (Schmidt TNS) that efficiently represents the Schmidt decomposition of finite- and even infinite-size quantum states with nontrivial bipartition boundary. The key idea is to represent the Schmidt coefficients (i.e., entanglement spectrum) and transformations in the decomposition to tensor networks (TNs) with linearly scaled complexity versus N. Specifically, the transformations are written as the TNs formed by local unitary tensors, and the Schmidt coefficients are encoded in a positive-definite matrix product state (MPS). Translational invariance can be imposed on the TNs and MPS for the infinite-size cases. The validity of Schmidt TNS is demonstrated by simulating the ground state of the quasi-one-dimensional spin model with geometrical frustration. Our results show that the MPS encoding the Schmidt coefficients is weakly entangled even when the entanglement entropy of the decomposed state is strong. This justifies the efficiency of using MPS to encode the Schmidt coefficients, and promises an exponential speedup on the full-state sampling tasks.
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 explore the relation between the entanglement of a pure state and its energy variance for a local one-dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance ${\ensuremath{\delta}}^{2}$ for NNN spins, with bond dimension scaling as $\sqrt{N}{D}_{0}^{1/\ensuremath{\delta}}$, where D0>1{D}_{0}>1D0​>1 is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.
We propose a way to construct a thermal pure quantum matrix product state (TPQ-MPS) that can simulate finite-temperature quantum many-body systems with a minimal numerical cost comparable to the matrix product algorithm for the ground state.The MPS was originally designed for the wave function with area law entanglement.However, by attaching the auxiliary sites to the edges of the random matrix product state, we find that the degree of entanglement is automatically tuned so as to recover the volume law of the entanglement entropy that characterizes the TPQ state.The finite-temperature physical quantities of the transverse Ising and the spin-1/2 Heisenberg chains evaluated by a TPQ-MPS show excellent agreement even for bond dimension ∼10-20 with those of the exact results.
We study thermal states of strongly interacting quantum spin chains and prove that those can be represented in terms of convex combinations of matrix product states. Apart from revealing new features of the entanglement structure of Gibbs states, our results provide a theoretical justification for the use of White's algorithm of minimally entangled typical thermal states. Furthermore, we shed new light on time dependent matrix product state algorithms which yield hydrodynamical descriptions of the underlying dynamics.