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arXiv:2310.16503 · Journal of Statistical Mechanics Theory and Experiment

Bootstrapping entanglement in quantum spin systems

Jiaju Zhang, Arash Jafarizadeh, M. A. Rajabpour

Published Jun 1, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract In this paper, we employ the bootstrap method, a technique that relies on consistency relations instead of direct diagonalization, to determine the expectation values in quantum many-body systems. We then use these values to assess the entanglement content of the system. Our work extends the bootstrap approach to quantum many-body systems, rather than single-body or few-body systems, concentrating on the well-known Lipkin–Meshkov–Glick (LMG) model with both transverse and longitudinal external magnetic fields. In the bootstrap method we solve the LMG model with up to 16 sites. Unlike previous studies that have focused mainly on ground-state properties, our methodology allows for the calculation of a broad range of properties, including energy spectrum, angular momentum, concurrence, tangle, residual tangle, and quantum Fisher information (QFI), for all eigenstates or a particular sector of the eigenstates, without referring to the explicit wavefunctions of these states. We show that this approach offers not only a new computational methodology but also a comprehensive view of both bipartite and multipartite entanglement properties across the entire spectrum of eigenstates. Specifically, we demonstrate that states typically found in the central region of the spectrum exhibit greater multipartite entanglement, as indicated by larger QFI values, compared to states at the edges of the spectrum. In contrast, concurrence displays the opposite trend. This observed behavior is in line with the monogamy principle governing quantum entanglement.

DOI
10.1088/1742-5468/addaa8

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Lipkin-Meshkov-Glick (LMG) models describe a set of NNN qubits which are located at the corners of an NNN-dimensional simplex and embedded in an external magnetic field. Because of its high coordinate number, quantum correlations in the models are quite nontrivial. For instance, in the large-NNN limit, two-site entanglement concurrence vanishes for any magnetic field. Multisite global entanglement and generalized global entanglement also tend to vanish in some quantum regions. We characterize the quantum correlations in the LMG model with multipartite nonlocality. In LMG models with anisotropy in the $x\ensuremath{-}y$ plane, for any fixed magnetic field, the ground-state nonlocality (denoted by S\mathcal{S}S) is found to scale as ${log}_{10}\mathcal{S}\ensuremath{\sim}aN+b$, with aaa and bbb the fitting parameters. For LMG models which are isotropic in the $x\ensuremath{-}y$ plane, nevertheless, the nonlocality for each Dicke state scales as ${log}_{10}\mathcal{S}\ensuremath{\sim}a\frac{1}{N}+b$, with b>0b>0b>0. Signals for the quantum phase transitions of the models will also be discussed. These results indicate that multipartite nonlocality captures some key ingredient of the ground-state quantum correlations in the models. In addition to the ground states, we also study multipartite nonlocality in the LMG models at finite temperatures. In the anisotropic models, the thermal stability of the nonlocality at low temperatures is determined by the energy gap. For the isotropic models, nevertheless, as the temperature increases, the nonlocality presents a sharp valley which is followed by a round peak. The behavior is far beyond the above-mentioned ``thermal stability vs energy gap'' picture. The underlying mechanics is the ``contribution inversion'' of low-lying energy states in the thermal-state nonlocality $\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{T})$; that is, in a low-temperature region, the first-excited state ``defeats'' the ground state and plays a dominant role in thermal-state nonlocality.
Simple physical interactions between spin-1∕21∕21∕2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-1∕21∕21∕2 particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-1∕21∕21∕2 particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an NNN-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the XXX-state). This XXX-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset AAA of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of N=2n+1N=2n+1N=2n+1 qubits is maximally entangled with the remaining qubits (set BBB) in the chain. In addition, we prove that by performing a local operation just on the subset BBB, one can transform the XXX-state into a direct product of kkk singlets shared by the parties AAA and BBB. This property of the XXX-state can be utilized for new secure multipartite communication protocols.
Entanglement microscopy reveals the true quantum correlations among the microscopic building blocks of many-body systems [Nat. Commun. 16, 96 (2025)]. Using this approach, we study the multipartite entanglement of the quantum Ising model in 1D, 2D, and 3D. We first obtain the full reduced density matrix (tomography) of subregions that have at most four sites via quantum Monte Carlo, exact diagonalization, and the exact solution in 1D. We then analyze both bipartite and genuine multipartite entanglement (GME) among the sites in the subregion. To do so, we use a variety of measures including the negativity, as well as a true measure of GME: the genuinely multipartite concurrence (or GME concurrence), and its computationally cheaper lower bound I2{I}_{2}I2​. We provide a complete proof that I2{I}_{2}I2​ bounds the GME concurrence, and show how the symmetries of the state simplify its evaluation. For adjacent sites, we find three- and four-spin GME present across large portions of the phase diagram, reaching a maximum near the quantum critical point. In 1D, we identify the singular scaling of the derivative dI2/dhd{I}_{2}/dhdI2​/dh approaching the critical point. We observe a sharp decrease in GME with increasing dimensionality, consistent with the monogamous nature of entanglement. Furthermore, we find that GME vanishes for subregions consisting of nonadjacent sites in both 2D and 3D, offering a stark illustration of the short-ranged nature of entanglement in equilibrium quantum matter [arXiv:2402.06677]. Finally, we analyze the most collective form of entanglement by evaluating the GME concurrence among all spins in the lattice, which can be obtained from a simple observable: the single-site transverse magnetization. This global concurrence is larger in 1D compared to 2D and 3D, but it is relatively less robust against perturbations such as local measurements.
Abstract The theory of multipartite entanglement provides a robust framework for understanding complex phenomena in quantum many-body systems, while multipartite entanglement also serves as a key resource in numerous quantum information processing tasks. Using density matrix renormalization group calculations, we study the multipartite entanglement in one-dimensional quantum long-range ferromagnetic and antiferromagnetic Ising chains with algebraically decaying interactions 1/ r α , respectively. To quantitatively probe the system’s multipartite entanglement structure, we employ the quantum Fisher information (QFI). Our results reveal two key behaviors: as the system size increases, the QFI rises, while the multipartite entanglement density decreases and converges toward a constant value. The latter trend points to a fundamental constraint on entanglement distribution in the thermodynamic limit. We further discuss the effects of the magnetic field and the exponent α on the QFI. We find that the peak values of the QFI increase with increasing α . Additionally, the discontinuity in the QFI can be regarded as a powerful tool for characterizing quantum phase transitions in the systems.
The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis {|j,m⟩}m=−jj in any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to 1/2 of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.
Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.