The area law of the bipartite information measure characterizes one of the most fundamental aspects of quantum many-body physics. In thermal equilibrium, the area law for the mutual information universally holds at arbitrary temperatures as long as the systems have short-range interactions. In systems with power-law decaying interactions, r^{-α} (r: distance), conditions for the thermal area law are elusive. In this Letter, we aim to clarify the optimal condition α>α_{c} such that the thermal area law universally holds. A standard approach to considering the conditions is to focus on the magnitude of the boundary interaction between two subsystems. However, we find here that the thermal area law is more robust than this conventional argument suggests. We show the optimal threshold for the thermal area law by α_{c}=(D+1)/2 (D: the spatial dimension of the lattice), assuming a power-law decay of the clustering for the bipartite correlations. Remarkably, this condition encompasses even the thermodynamically unstable regimes α<D. We verify this condition numerically, finding that it is qualitatively accurate for both integrable and nonintegrable systems. Unconditional proof of the thermal area law is possible by developing the power-law clustering theorem for α>D above a threshold temperature. Furthermore, the numerical calculation for the logarithmic negativity shows that the same criterion α>(D+1)/2 applies to the thermal area law for quantum entanglement.
The area law of the bipartite information measure characterizes one of the most fundamental aspects of quantum many-body physics. In thermal equilibrium, the area law for the mutual information universally holds at arbitrary temperatures as long as the systems have short-range interactions. In systems with power-law decaying interactions, r^{-α} (r: distance), conditions for the thermal area law are elusive. In this Letter, we aim to clarify the optimal condition α>α_{c} such that the thermal area law universally holds. A standard approach to considering the conditions is to focus on the magnitude of the boundary interaction between two subsystems. However, we find here that the thermal area law is more robust than this conventional argument suggests. We show the optimal threshold for the thermal area law by α_{c}=(D+1)/2 (D: the spatial dimension of the lattice), assuming a power-law decay of the clustering for the bipartite correlations. Remarkably, this condition encompasses even the thermodynamically unstable regimes α<D. We verify this condition numerically, finding that it is qualitatively accurate for both integrable and nonintegrable systems. Unconditional proof of the thermal area law is possible by developing the power-law clustering theorem for α>D above a threshold temperature. Furthermore, the numerical calculation for the logarithmic negativity shows that the same criterion α>(D+1)/2 applies to the thermal area law for quantum entanglement.
Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.
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.
We prove that the quantum Gibbs states of spin systems above a certain threshold temperature are approximate quantum Markov networks, meaning that the conditional mutual information decays rapidly with distance. We demonstrate the exponential decay for short-ranged interacting systems and power-law decay for long-ranged interacting systems. Consequently, we establish the efficiency of quantum Gibbs sampling algorithms, a strong version of the area law, the quasilocality of effective Hamiltonians on subsystems, a clustering theorem for mutual information, and a polynomial-time algorithm for classical Gibbs state simulations.
We present a general definition of quantum mutual entropy for infinitely extended quantum spin and fermion lattice systems. Using this, we establish a thermal area law in these infinitely extended quantum systems. The proof is based on the local thermodynamical stability (LTS), a variational principle in terms of the conditional free energy. Our thermal area law in quasi-local C*-systems applies to general interactions with well-defined surface energies. We also examine the quantum mutual entropy between the left- and right-sided infinite regions of one-dimensional lattice systems. For general translation-invariant finite-range interactions on such systems, the thermal equilibrium state at any temperature exhibits a finite mutual entropy between these infinite disjoint regions. This further implies that the infinitely large quantum entanglement characteristic of critical ground states in one-dimensional systems is drastically destroyed by even a small positive temperature.
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.
Many inexplicable phenomena in low-temperature many-body physics are a result of macroscopic quantum effects.Such macroscopic quantumness is often evaluated via long-range entanglement, that is, entanglement in the macroscopic length scale.Long-range entanglement is employed to characterize novel quantum phases and serves as a critical resource for quantum computation.However, the conditions under which long-range entanglement is stable, even at room temperatures, remain unclear.In this regard, this study demonstrates the unstable nature of bipartite long-range entanglement at arbitrary temperatures, which exponentially decays with distance.The proposed theorem is a no-go theorem pertaining to the existence of long-range entanglement.The obtained results are consistent with existing observations, indicating that long-range entanglement at nonzero temperatures can exist in topologically ordered phases, where tripartite correlations are dominant.The derivation in this study introduces a quantum correlation defined by the convex roof of the standard correlation function.Further, an exponential clustering theorem for generic quantum many-body systems under such a quantum correlation at arbitrary temperatures is established, which yields the primary result by relating quantum correlation with quantum entanglement.Moreover, a simple application of analytical techniques is demonstrated by deriving a general limit on the Wigner-Yanase-Dyson skew and quantum Fisher information; this is expected to attract significant attention in the field of quantum metrology.Notably, this study reveals the novel, general aspects of lowtemperature quantum physics and clarifies the characterization of long-range entanglement.