Quantum matter is weakly entangled at low energies

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 Rényi 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 Rényi 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.

• The paper establishes rigorous upper bounds on entanglement entropy for fixed-energy quantum states by mapping the problem to overlapping thermal subsystems.
• It demonstrates that reflection-symmetric and frustration-free systems exhibit area-law scaling through links between subsystem ground-state degeneracy and Schmidt rank.
• Numerical and analytical evidence from free fermion models and AKLT chains validate the proposed bounds and scaling laws for both gapped and gapless systems.

Weak Entanglement of Quantum Matter at Low Energies

Overview

The paper "Quantum matter is weakly entangled at low energies" (2604.14143) establishes rigorous upper bounds on the entanglement entropy of many-body quantum states with fixed energy expectation value for geometrically local Hamiltonians. Focusing on subsystems that comprise roughly half of the total system, the main result relates the maximum possible von Neumann and Rényi entanglement entropies to thermodynamic quantities, specifically the sum of thermal entropies in two fictitious subsystems at a temperature determined by the energy constraint. For frustration-free (FF) systems, these results directly link subsystem ground state degeneracies to Schmidt rank bounds, thereby proving area-law entanglement scaling independently of the spectral gap. The bounds are shown to be optimal up to subleading corrections for reflection-symmetric systems, and strong numerical and analytical evidence is provided for saturation in free fermion models and other physical settings.

Thermodynamic Mapping and Entanglement Bounds

The central technical innovation is the mapping of the entanglement maximization problem at fixed global energy to a classical thermodynamics problem. For a quantum system with finite-range interactions divided into subsystems $A$, $B$, and $C$ such that $A$ and $C$ do not interact directly (see (Figure 1)), the von Neumann entropy of $A$ for pure states with energy expectation $E$ is upper bounded by half the sum of the thermal entropies of two overlapping fictitious systems ($AB$ and $BC$) at temperature $T^*$, where $B$0 is set such that their energies sum to $B$1. This result generalizes to Rényi entropies for all indices.

Figure 1: Upper bound on entanglement entropy: For a quantum system partitioned into $B$2, $B$3, $B$4 with $B$5 and $B$6 not interacting, the entanglement entropy of $B$7 at fixed energy $B$8 is upper bounded by the sum of thermal entropies of fictitious $B$9 and $C$0 systems plus an area-law correction.

Furthermore, the bounds remain tight for reflection-symmetric systems, where the optimal upper bound equals the lower bound up to boundary corrections, proved by construction of lower-bound states that exploit boundary field configurations (Figure 2).

Figure 2: Lower bounds on maximum entanglement entropy in reflection-symmetric systems: Reflection symmetry enforces tightness of upper bounds on entanglement, as lower bounds derived from fixed boundary states converge to the same scaling.

Frustration-Free Systems and Area Laws

For FF Hamiltonians, the upper bound on ground-state Schmidt rank is set by the product of subsystem ground-state degeneracies, holding for both von Neumann and Rényi entropies and independently of the spectral gap. This analytical result implies that area-law entanglement in FF systems emerges whenever zero-temperature thermal entropies scale with the surface area rather than subsystem volume. The analysis is demonstrated concretely via examples such as AKLT and Motzkin chains, where subsystem Hamiltonians missing boundary terms exhibit exponential degeneracy growth with $C$1, guaranteeing area-law entanglement.

The construction applies directly to higher-dimensional FF systems, as exemplified by the division of a square lattice into $C$2, $C$3, $C$4 with local interactions (Figure 3).

Figure 3: Division of the square lattice into $C$5, $C$6, $C$7 for an FF Hamiltonian; missing terms at the entanglement cut lead to surface-area scaling of ground-state degeneracies and, consequently, area-law entanglement.

Entanglement Scaling at Nonzero Energies

For systems with conventional thermodynamic properties, the scaling of entanglement entropy at energies above the ground state is controlled by the behavior of the low-temperature specific heat capacity $C$8. The effective temperature $C$9 associated with the upper bound decreases with system size for subextensive energies. The paper derives explicit asymptotic scalings for entanglement entropy as functions of energy, system size, and $A$0:

• Gapped Systems ($A$1): Entanglement at energy $A$2 above ground state is upper bounded by $A$3, i.e., nearly area-law scaling with a logarithmic correction.
• Gapless Systems ($A$4): Entanglement at energy $A$5 scales as $A$6, e.g., $A$7 for metals.
• Disordered Systems ($A$8): Volume-law scaling up to a logarithmic reduction, i.e., $A$9.

These scalings are visualized in (Figure 4), linking thermodynamic behavior to entanglement bounds.

Figure 4: Relation between low-temperature specific heat $C$0 and the upper bounds on half-system entanglement entropy, for states at energy $C$1 above ground.

For extensive energies, the maximal entanglement is bounded by half the thermal entropy at the given energy, as saturated in chaotic eigenstates.

Numerical Evidence and Tightness of Bounds

The analytical scalings are validated numerically in free fermion systems, where maximal entanglement eigenstates at fixed subextensive energy display $C$2 scaling in $C$3 (Figure 5), consistent with predictions from thermal entropy bounds.

Figure 5: Comparison of upper bounds on entanglement (triangles) and eigenstate entanglement (circles) for free fermion systems in $C$4 and $C$5. Solid lines illustrate scaling $C$6, confirming tightness and optimality up to subleading corrections.

The paper further constructs lower bounds that match the upper bounds to leading order in $C$7 for reflection-symmetric geometries, demonstrating the optimality of the thermodynamic mapping.

Implications and Future Directions

The results formally connect thermodynamic properties of quantum matter (such as specific heat and ground-state degeneracy) to the maximal entanglement entropy achievable at fixed energy, for geometrically local systems. This establishes a direct constraint on simulation complexity for tensor network approaches and bounds on entanglement growth under Hamiltonian evolution. The independence from spectral gap in FF systems refines understanding of quantum computational complexity. The paper also links behavior of random eigenstates and pure states, providing a precise context for eigenstate thermalization in the low-energy regime.

The rigorous bounds derived raise several questions regarding simulation tractability for generic low-energy states, including the construction of states saturating the entanglement bound at energies $C$8 in $C$9 non-FF systems, and the role of quasiparticle configurational entropy. Investigating the dynamics of entanglement generation starting from low-energy states and its implications for physical observables remains relevant for future work.

Conclusion

This paper provides a comprehensive thermodynamic framework for bounding entanglement entropy in quantum matter at low energies. The mapping to overlapping fictitious thermal subsystems yields optimal, explicit, and universal scalings for entanglement, controlled by subsystem thermal entropy and ground-state degeneracy. The results have significant theoretical and practical implications for tensor network simulation, quantum complexity, and many-body dynamics, and delineate sharp constraints on entanglement growth for both FF and generic quantum systems.