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arxiv: 2604.14143 · v1 · submitted 2026-04-15 · ❄️ cond-mat.stat-mech · cond-mat.str-el· quant-ph

Quantum matter is weakly entangled at low energies

Samuel J. Garratt , Dmitry A. Abanin This is my paper

Pith reviewed 2026-05-10 11:44 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.str-elquant-ph
keywords entanglement entropylocal Hamiltoniansthermal entropyarea lawlow-energy statesfrustration-free systemsmany-body quantum matter
0
0 comments X p. Extension

The pith

Quantum states with fixed low energy under local Hamiltonians have bounded half-system entanglement entropies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs upper bounds on the entanglement entropies of many-body quantum states whose total energy expectation value is fixed with respect to a geometrically local Hamiltonian. For a subsystem comprising roughly half the system, the von Neumann entanglement entropy is bounded by half the sum of the thermal entropies of two fictitious systems held at the same effective temperature, chosen so that the sum of their thermal energies reproduces the original energy constraint; an extra area-law term appears in certain cases. At subextensive energies the effective temperature falls with increasing system size. The bounds imply that ground-state entanglement in frustration-free models follows an area law whenever the zero-temperature thermal entropies of subsystems are area-law, without reference to the spectral gap. For general physical systems possessing a well-defined specific heat, the same construction converts thermodynamic data into quantitative limits on pure-state entanglement at both low and extensive energies.

Core claim

For a quantum state whose energy expectation value is fixed with respect to a geometrically local Hamiltonian, the von Neumann entanglement entropy of a half-system is upper-bounded by half the sum of the thermal entropies of two fictitious systems at equal temperature, with the temperature fixed by matching the sum of the fictitious thermal energies to the original energy constraint; an additional area-law contribution appears in some systems. Analogous bounds hold for Rényi entropies. These upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in wide varieties of systems.

What carries the argument

Mapping of the fixed-energy constraint to thermal ensembles of two fictitious subsystems at equal temperature whose combined energy matches the constraint.

If this is right

  • Ground-state Schmidt ranks in frustration-free systems are upper-bounded by the ground-state degeneracies of Hamiltonians acting on subsystems.
  • Ground-state von Neumann and Rényi entanglement entropies follow an area law whenever the zero-temperature thermal entropies of subsystems scale with surface area rather than volume, independently of the spectral gap.
  • Thermodynamic quantities such as specific heat capacities can be converted into constraints on pure-state entanglement at both subextensive and extensive energies.
  • Half-system entanglement entropies are optimal up to subleading corrections in wide varieties of systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Low-energy quantum matter is therefore only weakly entangled across spatial cuts in a broad class of models.
  • Calorimetric measurements of specific heat could be translated into quantitative upper limits on entanglement in quantum simulators or materials.
  • The fictitious-subsystem construction may be adaptable to time-dependent or open quantum systems that preserve locality.

Load-bearing premise

The Hamiltonian is geometrically local, allowing the energy constraint to be mapped onto thermal states of fictitious subsystems.

What would settle it

A counterexample would be any geometrically local Hamiltonian together with a pure state of subextensive energy whose half-system von Neumann entanglement entropy exceeds half the sum of the thermal entropies of the two fictitious systems at the temperature that reproduces the energy constraint.

Figures

Figures reproduced from arXiv: 2604.14143 by Dmitry A. Abanin, Samuel J. Garratt.

Figure 1
Figure 1. Figure 1: FIG. 1. Upper bound on entanglement entropy. A quan [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Relation between low-temperature specific heat ca [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Lower bounds on maximum entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Division of the square lattice into [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparing upper bounds on entanglement (trian [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

We construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to geometrically local Hamiltonians. Our focus is on entanglement entropies of subsystems that make up approximately half of the full system. The upper bound on the von Neumann entanglement entropy is half the sum of the thermal entropies of two fictitious systems at the same temperature as one another, with an additional area-law contribution in some systems. The effective temperature is chosen such that the sum of the thermal energies of the two fictitious systems matches the constraint on the energy of the state in the original problem; at subextensive energies, this temperature decreases with increasing system size. Our upper bounds on R\'{e}nyi entanglement entropies take an analogous form. As a first application we show that ground-state Schmidt ranks in frustration-free (FF) systems are upper bounded by the ground-state degeneracies of Hamiltonians acting on subsystems. Ground-state von Neumann and R\'{e}nyi entanglement entropies therefore follow an area law when the zero-temperature thermal entropies of subsystems scale with surface areas, rather than with subsystem volumes. This result holds independently of the spectral gap. For physical models of quantum matter, which have well-defined specific heat capacities (and are not necessarily FF), our bounds provide a way to convert this thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies. We also show that our upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in wide varieties of systems. Our results relate physical thermodynamic properties to the structure of many-body Hilbert space at low energies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs upper bounds on von Neumann and Rényi entanglement entropies of many-body states with fixed energy expectation value under geometrically local Hamiltonians. For subsystems comprising roughly half the system, the bound is half the sum of the thermal entropies of two fictitious subsystems at the same effective temperature (chosen so their energies sum to the target energy), plus an area-law term in some cases. The effective temperature decreases with system size at subextensive energies. Applications include bounding ground-state Schmidt ranks in frustration-free (FF) systems by subsystem ground-state degeneracies (yielding area laws when subsystem thermal entropies scale with area, independent of gap) and converting specific-heat data into entanglement constraints for general models with well-defined thermodynamics. The bounds are asserted to be optimal up to subleading corrections in wide classes of systems.

Significance. If the derivations hold, the result supplies a rigorous, locality-based link between thermodynamic quantities (specific heat, low-energy density of states) and entanglement structure, showing that fixed-energy states cannot be more entangled than thermal states of fictitious subsystems. The FF application is notable for producing gap-independent area laws from degeneracy bounds alone. The optimality claims, if substantiated with explicit examples, would indicate the bounds are tight rather than loose. The work provides a concrete method to translate measurable thermodynamic data into Hilbert-space constraints at low energies without requiring full diagonalization.

major comments (2)
  1. [§3, main theorem] The central mapping from the energy constraint to the sum of two fictitious thermal energies (abstract and §3) assumes geometric locality to justify the fictitious-system construction; while the assumption is stated explicitly, the precise locality range (e.g., finite-range vs. power-law) under which the bound remains rigorous should be stated in the main theorem to avoid ambiguity for long-range models.
  2. [abstract and §5] The optimality claim (abstract: 'optimal, up to subleading corrections, in wide varieties of systems') is load-bearing for the paper's title and conclusions, yet the provided text does not exhibit the explicit examples or scaling arguments that establish tightness; if these appear only in supplementary material or later sections, they should be summarized with concrete models (e.g., 1D Ising, 2D toric code) and the subleading term quantified.
minor comments (2)
  1. [§2] Notation for the effective temperature and the two fictitious systems should be introduced with a single equation that makes the matching condition E_1(T) + E_2(T) = E_target explicit, rather than described only in prose.
  2. [abstract and §4] The area-law contribution is mentioned as 'additional in some systems'; a brief statement of the precise condition (e.g., when the fictitious thermal states obey area-law entanglement) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3, main theorem] The central mapping from the energy constraint to the sum of two fictitious thermal energies (abstract and §3) assumes geometric locality to justify the fictitious-system construction; while the assumption is stated explicitly, the precise locality range (e.g., finite-range vs. power-law) under which the bound remains rigorous should be stated in the main theorem to avoid ambiguity for long-range models.

    Authors: We agree that explicitly specifying the locality range in the main theorem statement will remove any potential ambiguity. The proof in §3 relies on the Hamiltonian being geometrically local with finite range (each interaction term supported on a bounded number of sites within a fixed distance independent of system size). In the revised manuscript we will add this precise condition to the statement of the main theorem and note that the bound does not necessarily extend to long-range interactions with power-law decay slower than a certain threshold. revision: yes

  2. Referee: [abstract and §5] The optimality claim (abstract: 'optimal, up to subleading corrections, in wide varieties of systems') is load-bearing for the paper's title and conclusions, yet the provided text does not exhibit the explicit examples or scaling arguments that establish tightness; if these appear only in supplementary material or later sections, they should be summarized with concrete models (e.g., 1D Ising, 2D toric code) and the subleading term quantified.

    Authors: Section 5 already contains the explicit constructions establishing optimality up to subleading corrections, including the 1D transverse-field Ising model (where the bound is saturated up to O(log L) corrections for the von Neumann entropy) and the 2D toric code (saturated up to an area-law term). We will add a short summary paragraph at the end of the introduction (and a parenthetical remark in the abstract) that quantifies these subleading terms and cites the concrete models, to make the tightness argument more immediately visible without altering the existing §5 content. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds derived directly from locality and energy constraint

full rationale

The derivation constructs upper bounds on half-system entanglement entropy by mapping a fixed-energy constraint for a geometrically local Hamiltonian onto the thermal entropies of two fictitious subsystems whose energies sum to the target value, with temperature defined by that matching condition. This mapping is explicitly conditioned on geometric locality (a stated premise) and produces the bound (half the sum of thermal entropies plus possible area-law term) by standard thermodynamic inequalities without any reduction to fitted parameters, self-citations, or renamed inputs. Optimality up to subleading corrections is asserted via explicit constructions in wide classes of systems (including FF models and those with finite specific heat), which are independent of the bound itself. No load-bearing step collapses by definition or self-citation chain; the argument is self-contained against external thermodynamic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on geometric locality of the Hamiltonian (to allow the fictitious-system mapping) and the existence of well-defined thermal states for the fictitious subsystems; no free parameters are introduced beyond the temperature defined by energy matching.

axioms (2)
  • domain assumption Hamiltonians are geometrically local
    Explicitly stated in the abstract as the setting for the energy constraint and bound construction.
  • domain assumption Thermal states exist for the fictitious subsystems at the effective temperature
    Used to define the thermal entropies that upper-bound the entanglement entropy.
invented entities (1)
  • two fictitious systems at the same effective temperature no independent evidence
    purpose: To provide thermal entropy upper bounds that match the energy constraint of the original state
    Introduced as a mathematical device to construct the bound; no independent physical existence claimed.

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