The area law for entanglement provides one of the most important connections between information theory and quantum many-body physics. It is not only related to the universality of quantum phases, but also to efficient numerical simulations in the ground state. Various numerical observations have led to a strong belief that the area law is true for every non-critical phase in short-range interacting systems. However, the area law for long-range interacting systems is still elusive, as the long-range interaction results in correlation patterns similar to those in critical phases. Here, we show that for generic non-critical one-dimensional ground states with locally bounded Hamiltonians, the area law robustly holds without any corrections, even under long-range interactions. Our result guarantees an efficient description of ground states by the matrix-product state in experimentally relevant long-range systems, which justifies the density-matrix renormalization algorithm.
The area law for entanglement provides one of the most important connections between information theory and quantum many-body physics. It is not only related to the universality of quantum phases, but also to efficient numerical simulations in the ground state. Various numerical observations have led to a strong belief that the area law is true for every non-critical phase in short-range interacting systems. However, the area law for long-range interacting systems is still elusive, as the long-range interaction results in correlation patterns similar to those in critical phases. Here, we show that for generic non-critical one-dimensional ground states with locally bounded Hamiltonians, the area law robustly holds without any corrections, even under long-range interactions. Our result guarantees an efficient description of ground states by the matrix-product state in experimentally relevant long-range systems, which justifies the density-matrix renormalization algorithm.
Abstract The area law for entanglement provides one of the most important connections between information theory and quantum many-body physics. It is not only related to the universality of quantum phases, but also to efficient numerical simulations in the ground state. Various numerical observations have led to a strong belief that the area law is true for every non-critical phase in short-range interacting systems. However, the area law for long-range interacting systems is still elusive, as the long-range interaction results in correlation patterns similar to those in critical phases. Here, we show that for generic non-critical one-dimensional ground states with locally bounded Hamiltonians, the area law robustly holds without any corrections, even under long-range interactions. Our result guarantees an efficient description of ground states by the matrix-product state in experimentally relevant long-range systems, which justifies the density-matrix renormalization algorithm.
The entanglement area law is a universal principle that characterizes quantum many-body phases and underpins tensor network algorithms. Traditionally, its validity has been limited to systems with short-range interactions and bounded local energy. Achieving a complete generalization that removes both of these constraints has been a longstanding goal in quantum many-body theory, especially for interacting boson systems where unbounded energy presents intrinsic difficulties. In this work, we rigorously prove the area law for one-dimensional interacting boson systems with long-range interactions, covering broad models including the Bose-Hubbard and ϕ4 classes. Furthermore, we establish an efficiency guarantee for Matrix-Product-State approximations of the ground states, offering a practical route to numerical simulation. One of our main technical contributions is a general method for Hilbert space dimension reduction, whose applicability extends to arbitrary spatial dimensions. These results address two major challenges simultaneously and provide important foundations for simulating long-range cold atomic systems. Entanglement area law has been established in systems with short range interactions and bounded local energy. Here the authors extend the proof to 1D bosonic systems with long-range interactions and unbounded local energy and outline implications for matrix product state approximations of the ground states.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.
The Rényi area law conjecture is false below α=1 in high dimensions
Area laws express the expectation that entanglement in gapped quantum ground states scales with a boundary rather than a volume. In one dimension this holds for every Rényi entropy, and the Rényi area-law conjecture asserts the same in higher dimensions for each fixed 0<α<1. Here we disprove this conjecture in every spatial dimension two and higher. We construct finite-range Hamiltonians with fixed local dimension, bounded interaction strength, unique ground states and a uniform spectral gap, whose ground states exhibit volume-law Rényi entanglement while obeying a von Neumann area law and remaining connected to product states through uniformly gapped local paths. The mechanism is error-space complexity: locally identifiable components of arbitrarily small weight can encode states with high entanglement complexity. We also establish the worst-case rank-error trade-off for approximate ground-state projectors, a key tool in one-dimensional area-law proofs. At fixed gap, the optimal Schmidt rank at projection error ε across a planar cut of cross-section A scales as exp[Θ~((Alogε−1)2/3)] when logε−1≳A. Thus α=1 is the sharp threshold for a universal Rényi area law within this adiabatically connected class.
We prove that the entanglement entropy of any state evolved under an arbitrary 1/r^{α} long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α>D+1. We also prove that for any α>2D+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.