Quantum states weakly linked at low energy levels

Samuel J. Garratt and Dmitry A. Abanin of Princeton University, along with colleagues at École Polytechnique Fédérale de Lausanne and Google Research, have established new limits on the entanglement of quantum states. Their work reveals that entanglement entropies of subsystems comprising approximately half of a larger system are bounded by a formula linking them to the thermal entropies of two fictitious systems at the same temperature.

The researchers demonstrate that ground states with zero-temperature thermal entropies scaling with surface areas, rather than volume, are demonstrably area-law entangled. These findings provide a pathway to translate thermodynamic data into constraints on pure-state entanglement for models of quantum matter at both low and higher energies.

Entanglement Bounds Constrain Simulations of Quantum Matter

Upper bounds on entanglement entropies directly link the complexity of quantum states to their energy levels, a novel way to assess the resources needed to simulate quantum materials, particularly at low energies. The established bounds decrease in effectiveness as energy levels fall below a certain threshold, with the effective temperature of the fictitious systems diminishing as system size increases. This suggests a unique interplay between energy scale, system dimensions and the limits of quantum entanglement, potentially impacting the feasibility of simulating complex quantum phenomena.

These findings are particularly relevant to simulations of zero-temperature response functions, where energies are typically of order unity, and have implications for the efficiency of tensor-network simulations. While some local Hamiltonians are known to generate highly entangled ground states that violate the area law, the new constraints identify properties of quantum matter that limit simulation complexity. The authors note that the bounds can help refine computational resource estimates for modeling these phases and understanding their universal properties.

Thermal Entropy Relationship to Subsystem Entanglement

The research demonstrates a relationship between maximizing subsystem entanglement with a fixed global energy and maximizing thermal entropy in two independent systems constrained by a combined energy limit. This connection manifests in a theorem stating that the maximum entanglement entropy is less than or equal to the sum of thermal entropies from two systems, each with a temperature chosen to match the original system’s energy constraint.

The effective temperatures used in these calculations decrease as system size increases at subextensive energies. The bounds developed also offer insights into the computational resources needed to simulate quantum materials.

Von Neumann and Rényi Entanglement Entropy Inequalities

The maximum possible von Neumann entanglement entropy of a subsystem A within a pure state is bounded by half the sum of thermal entropies, calculated for two systems with specific structures, and an additional area-law contribution in certain cases. These thermal entropies are not those of a standard Gibbs state, but rather maximum values attained subject to a fixed energy constraint. The formalism developed also allows for the conversion of thermodynamic data, specifically specific heat capacities, into constraints on pure-state entanglement at both subextensive and extensive energies.

The researchers state that this broadens the applicability to models of quantum matter. The bounds developed offer a way to understand entanglement in systems with well-defined heat capacities, even those that are not necessarily frustration-free.

Area Law Entanglement in Frustration-Free Systems

Entanglement entropies of subsystems are demonstrably bounded by a system’s energy scale, with the precision of that bound dependent on whether the system is frustration-free, according to new work. For physical models of quantum matter, the derived bounds provide a way to convert thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies.

Specific Heat Capacity as a Constraint on Entanglement

Low-temperature specific heat capacity, when scaling as c(T) = o(T^(1/(d-1))), limits the entanglement contribution from excitations to an area law of $L^{d–1}$. This means excitation-driven entanglement is often secondary to ground-state entanglement, a relationship the work anticipates will aid in understanding the complexity of tensor network representations of low-energy states. Efficient tensor network representation of ground states in frustration-free Hamiltonians is already known, and these results now constrain the entanglement of their excitations, offering a more complete picture of quantum system behavior.

Optimality of Half-System Entanglement Upper Bounds

The work demonstrates that, in many frustration-free systems spanning two or more dimensions, the contribution of excited states to von Neumann entanglement entropy is secondary to that of the ground state, simplifying the analysis of complex quantum systems. This temperature, denoted as $T^{\text{*}}$, is determined by a condition ensuring the combined thermal energies of the fictitious systems match the energy constraint of the original quantum state.

Analogous upper bounds also apply to Rényi entanglement entropies, broadening the scope of this analytical approach. For frustration-free ground states, the derived inequalities provide simple upper bounds on entanglement entropies, offering a direct link between system properties and entanglement behavior.

Ground State Schmidt Rank in Frustration-Free Systems

This connection between energy scale and entanglement is central to characterizing quantum states in frustration-free systems. The derived inequalities apply universally, encompassing scenarios where the ground state minimizes energy for every local Hamiltonian term. The work demonstrates that exceeding area-law scaling for ground-state entanglement requires ground state degeneracies of subsystems to grow faster than exponentially with surface area. In systems with well-defined specific heat capacities and subextensive energies, the density of states at low energy governs the scaling of upper bounds on entanglement entropy.

Low-Energy Entanglement and Response Functions

This refinement clarifies the relationship between system characteristics and the maximum achievable entanglement, offering a more precise understanding of quantum state complexity. This result has implications for understanding the limits of tensor network simulations, as highly entangled ground states, those violating the area law of subsystem entanglement entropy by a power of system size, appear to preclude efficient representation using these methods. Identifying the physical properties that constrain simulation complexity, both in ground states and low-energy excited states, is essential for predicting response functions.

The analysis also provides a pathway to constrain entanglement in quantum states relevant to simulating zero-temperature response functions, offering computational resource limits for tensor-network representations. This is especially valuable for phases with frustration-free Hamiltonian representatives, such as quantum ferromagnets, valence-bond solids and those found in the fractional quantum Hall effect. “Since our result bounds the amount of entanglement that can be generated in simulations of zero-temperature response, it constrains the computational resources required for tensor-network representations of quantum states,” the researchers state, highlighting the practical implications of their work.

Subextensive Energies and Temperature Dependence

This means larger systems exhibit a more constrained capacity for entanglement when considering energies that don’t scale with system volume. “At subextensive energies, this temperature decreases with increasing system size,” the paper notes, highlighting a unique relationship between energy scale, system size and the ultimate limits of quantum entanglement in these complex systems.

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