Quantum matter is weakly entangled at low energies

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.