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arXiv:cond-mat/0512558

Entanglement Properties Of Quantum Many-Body Wave Functions

J. W. Clark, Aikaterini Mandilara, MANFRED L. RISTIG

Published Dec 21, 2005• 1 citation
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Abstract

Entanglement properties of the trial many-body wave functions in variational treatments of the transverse Ising model in two, three, and four dimensions are investigated. Based on data for magnetizations and correlation functions generated by for symmetry-breaking Hartree-Jastrow trial ground states, results are presented for several measures of bipartite entanglement, including the von Neumann entropy of a single spin in the lattice, the concurrence between two spins, and a lower bound on the localizable entanglement, as functions of coupling strength. A simple relation is established between the Hartree-Jastrow states and the nilpotent polynomial representation of multipartite entanglement. In addition, some general comments are made on the entanglement properties of classes of many-body wave functions commonly adopted in theories of strongly correlated quantum systems.

DOI
10.48550/arxiv.cond-mat/0512558

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Entanglement properties of the trial many-body wave functions in variational treatments of the transverse Ising model in two, three, and four dimensions are investigated. Based on data for magnetizations and correlation functions generated by for symmetry-breaking Hartree-Jastrow trial ground states, results are presented for several measures of bipartite entanglement, including the von Neumann entropy of a single spin in the lattice, the concurrence between two spins, and a lower bound on the localizable entanglement, as functions of coupling strength. A simple relation is established between the Hartree-Jastrow states and the nilpotent polynomial representation of multipartite entanglement. In addition, some general comments are made on the entanglement properties of classes of many-body wave functions commonly adopted in theories of strongly correlated quantum systems.
The entanglement properties of correlated wave functions commonly employed in theories of strongly correlated many-body systems are studied. The variational treatment of the transverse Ising model within correlated-basis theory is reviewed, and existing calculations of the one- and two-body reduced density matrices are used to evaluate or estimate established measures of bipartite entanglement, including the Von Neumann entropy, the concurrence, and localizable entanglement, for square, cubic, and hypercubic lattice systems. The results discussed in relation to the findings of previous studies that explore the relationship of entanglement behaviors to quantum critical phenomena and quantum phase transitions. It is emphasized that Jastrow-correlated wave functions and their extensions contain multipartite entanglement to all orders.
This thesis is devoted to the theoretical characterization and generation of strongly correlated many-body states from the perspective of quantum information theory.We first study the entanglement properties of general many-body states and define a new measure for multipartite entanglement, the "Localizable Entanglement (LE)".It quantifies the maximal entanglement that can be localized, on average, between two parties by performing local measurements on the other parties.This measure provides knowledge of the "entanglement length", which is of particular importance in the context of quantum repeaters for long-distance quantum communication.We prove upper and lower bounds on LE in terms of two-body correlation functions and develop a method for the numerical computation of LE in one-dimensional spin systems.We analyze the localizable entanglement in various spin systems and observe characteristic features for a quantum phase transition such as a diverging entanglement length.In the second part of the thesis we focus on a specific many-body system: ultra-cold atoms in an optical lattice.This system is a promising candidate for the first physical realization of a "quantum simulator", but the temperature in current experiments is still too high.We propose, analyze and compare several schemes to cool the atoms close to the ground state of the strong interaction regime.In particular, we devise an algorithmic cooling protocol that combines occupation number filtering with spin-dependent lattice shifts.In addition, we propose two different physical realizations of filtering and also design protocols that generate an ensemble of quantum registers for quantum computation.In the third part we propose how to create entanglement in small atomic clouds by deforming and rotating the harmonic trapping potential.The resulting states are entangled in their motional degrees of freedom and can be identified with strongly correlated fractional quantum Hall states.For the case of two, three and four atoms we show how to adiabatically transform the unentangled ground state into the desired entangled state.We further discuss characteristic features of these states and propose how to create and detect them experimentally using an optical lattice setup. A Proof of the lower bound on LEA.1 Theorem on connected correlation functions and local measurements . . . . . . . . .
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 We present a method to measure the von Neumann entanglement entropy of ground states of quantum many-body systems which does not require access to the system wave function. The technique is based on a direct thermodynamic study of lattice entanglement Hamiltonians—recently proposed in the paper [Dalmonte et al 2018 Nat. Phys. 14 827] via field theoretical insights—and can be performed by quantum Monte Carlo methods. We benchmark our technique on critical quantum spin chains, and apply it to several two-dimensional quantum magnets, where we are able to unambiguously determine the onset of area law in the entanglement entropy, the number of Goldstone bosons, and to check a recent conjecture on geometric entanglement contribution at critical points described by strongly coupled field theories. The protocol can also be adapted to measure entanglement in experiments via quantum quenches.
We study the bipartite entanglement per bond to determine characteristic features of the phase diagram of various quantum spin models in different spatial dimensions. The bipartite entanglement is obtained from a tensor network representation of the ground state wave-function. Three spin-1/2 models (Ising, XY, XXZ, all in a transverse field) are investigated. Imaginary-time evolution (TEBD in 1D, `simple update' in 2D and 3D) is used to determine the ground states of these models. The phase structure of the models is discussed for all three dimensions.
THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.