Entanglement Area Laws for Long-Range Interacting Systems

Abstract

We prove that the entanglement entropy of any state evolved under an arbitrary 1=r α long-rangeinteracting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D þ 1. We also prove that for any α > 2D þ 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions. DOI: 10.1103/PhysRevLett.119.050501 Quantum many-body systems often have approximately local interactions, and this locality has profound effects on the entanglement properties of both ground states and the states created dynamically after a quantum quench. For example, the entanglement entropy, defined as the entropy of the reduced state of a subregion, often scales as the boundary area of the subregion for ground states of shortrange interacting Hamiltonians [1]. This "area law" of entanglement entropy is in sharp contrast to the behavior of thermodynamic entropy, which typically scales as the volume of the system. While the study of area laws originates from black hole physics [2,3], area laws have received considerable attention recently in the fields of quantum information and condensed matter physics. In particular, area laws are known to be closely related to the velocity of information propagation in quantum lattices [4], quantum critical phenomena [5], bulk-boundary correspondence [6], efficient classical simulation of quantum systems [7], topological order [8], and many-body localization [9].However, the description of many-body systems in terms of local interactions is often only an approximation, and not always a good one; in numerous systems of current interest, ranging from frustrated magnets and spin glasses [10,11] to atomic, molecular, and optical systems [12][13][14][15][16][17], longrangeinteractions are ubiquitous and lead to qualitatively new physics, e.g., giving rise to novel quantum phases and dynamical behaviors [18][19][20][21][22][23][24][25], and enabling speedups in quantum information processing [26][27][28][29][30]. Particles in these systems generally experience interactions that decay algebraically (∼1=r α ) in the distance (r) between them. As might be expected, α controls the extent to which the system respects notions of locality developed for short-range interacting systems: For α sufficiently small, it is well established [19] that locality may be completely lost, and for α sufficiently large there is ample numerical and analytical evidence [31][32][33][34] that area laws may persist. However, there is currently no general and rigorous understanding of when area laws do or do not survive the presence of long-range interactions.The modern understanding of area laws draws heavily from several rigor...

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“…This section includes the numerical performance of the fourth-order product formula (PF4) used to simulate the evolution of the system given in Eq. (25) for time T = n. We plot this numerical performance as well as the theoretical estimates for the gate counts of the PF4, LCU, QSP, and HHKL algorithms in Fig. 9.…”

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