Quantum matter is weakly entangled at low energies

The gist The authors construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to geometrically local Hamiltonians, relating this to constraints in statistical mechanics.

Thermodynamic Mapping for Entanglement Bounds

The central result establishes a relation between maximizing subsystem von Neumann entanglement entropy subject to a global energy constraint and maximizing the total thermal entropy of two independent systems at the same temperature Our central result is a relation between a problem in quantum information theory, the maximization of subsystem von Neumann entanglement entropy subject to a constraint on the global energy expectation value, and a classic problem in statistical physics, in which the total thermal entropy of two independent systems is maximized subject to a constraint on the sum of their energies. This relation leads to an upper bound on subsystem entanglement that is useful for subsystems which make up approximately half of the full system, and that is expressed in terms of subsystem heat capacities.

Upper Bounds for von Neumann and R´enyi Entropies

The authors construct analogous upper bounds on R´enyi entanglement entropies. For pure states, they derive Corollary 1, which states that the maximum possible value of the half-system entanglement entropy is bounded by half the sum of thermal entropies Our upper bound on half-system (von Neumann and R´enyi) entanglement entropies are optimal up to subleading corrections in wide varieties of systems. For general R´enyi indices α, the result is given by Theorem 2, which states that the maximum possible value of SA,α(ψ) + SC,α(ψ) is upper bounded by a sum involving thermal entropies.

Ground-State Entanglement in Frustration-Free Systems

For frustration-free (FF) systems with Hamiltonians of the form H = Pj hj, the authors show that ground-state Schmidt ranks are upper bounded by products of ground-state degeneracies of subsystems. If zero-temperature thermal entropies (i.e., logarithms of ground-state degeneracies) of subsystem Hamiltonians are upper bounded by subsystem surface areas, then ground states are area-law entangled. This result holds independently of the spectral gap.

Scaling with Energy and System Size

For gapped systems, at subextensive energies, the half-system entanglement entropy is upper bounded by (i) the product of the total energy, in units of the gap, and (ii) a logarithm of system volume (up to a prefactor). For gapless systems whose heat capacities scale as powers of the temperature, they establish scaling for the upper bound with both energy and system size. In disordered systems whose heat capacities scale inversely with powers of logarithms of the temperature, at subextensive energies, their upper bound scales with volume up to a multiplicative logarithmic reduction.

Nonzero Energy Entanglement in Generic Systems

For states at nonzero energy E − EH 0 above the ground state, they show that for systems whose heat capacities are self-averaging, the maximum possible entanglement entropy is upper bounded by half the thermal entropy associated with that energy E. For gapped systems where E − EH 0 = O(Lα) with d − 1 ≤ α < d, the entanglement entropy is upper bounded as SA(ψ) = O L(d−1 lnL). For power-law heat capacity systems, the bound scales as SA(ψ) = O L(d+αγ)/(1+γ).

Lower Bounds and Optimality

In Appendix A, they derive a simple lower bound on the maximum possible values of SA,α(ψ) for states with ⟨ψHψ⟩ = E, which shows that for E = EH 0 + Ω(L(d−1)) our upper bounds are optimal up to corrections that are subleading at large L in systems with conventional thermodynamic properties. The lower bound construction uses a specific choice of pure state and thermal states in fictitious thermodynamics problems.

Entanglement Dynamics and Interpretation

The authors conclude that their bounds provide insights into the complexity of simulating time evolution on classical computers, as entanglement is related to the computational resources required to represent quantum states as tensor networks. They show that for FF systems, if the zero-temperature thermal entropy of a subsystem grows with its surface area but not with its volume, then ground states of the full system are area-law entangled.

Application to Free Fermions

In Appendix C, they numerically show that even in non-interacting systems of fermions, there exist families of low-energy eigenstates whose entanglement entropies follow the scaling predicted in their upper bound (53), although with a modified prefactor. This implies that the scaling identified in (53) can be attributed to the presence of large numbers of long-wavelength excitations rather than with states whose mode occupation numbers exhibit large quantum fluctuations.

1. Enhanced Tensor Network Simulation Complexity Assessment: The improved system can determine the complexity of simulating time evolution on classical computers by using entanglement bounds to constrain required tensor network resources for geometrically local Hamiltonians (Sec. I, Sec. VI).

2. Ground State Complexity Prediction for Frustration-Free Systems: The AI can establish that ground-state Schmidt ranks of FF systems are upper bounded by products of ground-state degeneracies of subsystems (Sec. IV A), allowing for the prediction of whether a specific class of quantum matter will exhibit area-law entanglement based on local Hamiltonian properties.

3. Real-Time Entanglement Dynamics Forecasting: The system can predict how quenches from weakly entangled low-energy states (in particular, states with subextensive energies) lead to the emergence of maximally entangled states on physical time scales by leveraging the relationship between thermal entropy and entanglement growth (Sec. VI).

4. Thermodynamic Constraint Mapping for Generic States: The AI can convert thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies for physical models with defined specific heat capacities, providing bounds based on the total thermal entropy of two independent systems at the same temperature as one another (Sec. I).

5. Subextensive Energy Entanglement Scaling: The improved system can predict that for gapped systems, the half-system entanglement entropy is upper bounded by (i) the product of the total energy, in units of the gap, and (ii) a logarithm of system volume at subextensive energies (Sec. V D 1), enabling accurate complexity prediction for low-energy states.

6. Non-FF System Entanglement Bounds: The AI can calculate the upper bound on entanglement for gapped systems at energy scaling as E − EH0 = O(Lα) where d - 1 ≤ α < d, yielding a result of SA(ψ) = O L(d-1) lnL (Sec. V D 3), providing a precise measure of complexity for general gapped systems.