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A topological signature of multipartite entanglement

Fabio Lingua, Wei Wang, Liana Shpani, Barbara Capogrosso-Sansone

Published May 17, 2019• 0 citations
OpenAlex

Abstract

In this manuscript, we present a proposal to relate topological structure of worldline configurations to multipartite entanglement. Configurations result from the path-integral formulation of the density matrix in the limit of zero temperature. We consider hard-core bosons for which configurations, i.e. collections of particle paths, can be seen as geometric braids with a certain topological structure. We propose that properties of worldline configurations may realize a comprehensive deciphering of multipartite entanglement. By means of Monte Carlo calculations, we study checkerboard, stripe, valence-bond solids, $\mathbb{Z}_2$ topologically ordered spin liquid, and superfluid phase. We find that each ground-state is characterized by a certain `topological spectrum' which can be used to differentiate among different ground-states.

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In this manuscript, we present a proposal to relate topological structure of worldline configurations to multipartite entanglement. Configurations result from the path-integral formulation of the density matrix in the limit of zero temperature. We consider hard-core bosons for which configurations, i.e. collections of particle paths, can be seen as geometric braids with a certain topological structure. We propose that properties of worldline configurations may realize a comprehensive deciphering of multipartite entanglement. By means of Monte Carlo calculations, we study checkerboard, stripe, valence-bond solids, Z2\mathbb{Z}_2Z2​ topologically ordered spin liquid, and superfluid phase. We find that each ground-state is characterized by a certain `topological spectrum' which can be used to differentiate among different ground-states.
Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular S\mathcal{S}S and U\mathcal{U}U matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular S\mathcal{S}S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.
In this manuscript, we study braiding properties of worldline configurations for a variety of ground-states of hardcore Bose-Hubbard models in two dimensions. Configurations are collections of particle paths and result from the path-integral formulation of statistical mechanics. For hard-core bosons, configurations can be seen as geometric braids and therefore can be assigned a certain topological structure, i.e. a way to classify braiding events among worldlines. By means of Monte Carlo calculations, we study superfluid phase and a variety of insulating phases and observe that ground-states of different quantum phases correspond to different probability distributions of braiding properties.
We demonstrate that multipartite entanglement is able to characterize one-dimensional topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the original and dual lattices. We investigate an exactly-solvable long-range extended spin-12\frac{1}{2}21​ Ising model with nontrivial Z\mathbf{Z}Z-symmetry topological order identified equivalently by winding numbers and paired Majorana zero modes at each end. For topologically ordered phases with high winding numbers, we can define a Z\mathbf{Z}Z topological invariant by the scaling coefficient of the dual quantum Fisher information density. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topologically ordered state has promising applications in robust quantum computation and quantum metrology, and could be generalized to higher-dimensional systems.
A useful approach to characterize and identify quantum phase transitions lies in the concept of multipartite entanglement. In this paper, we consider well-known measures of multipartite (global) entanglement, i.e., average linear entropy of one-qubit and two-qubit reduced density matrices, in order to study topological quantum phase transition (TQPT) in the Kitaev Toric code Hamiltonian with a nonlinear perturbation. We provide an exact mapping from aforementioned measures in the above model to internal energy and energy-energy correlations in the classical Ising model. Accordingly, we find that the global entanglement shows a continuous and sharp transition from a maximum value in the topological phase to zero in the magnetized phase in a sense that its first-order derivative diverges at the transition point. In this regard, we conclude that not only can the global entanglement serve as a reasonable tool to probe quantum criticality at TQPTs, but it also can reveal the highly entangled nature of topological phases. Furthermore, we also introduce a conditional version of global entanglement which becomes maximum at the critical point. Therefore, regarding a general expectation that multipartite entanglement reaches maximum value at the critical point of quantum many-body systems, our result proposes that the conditional global entanglement can be a good measure of multipartite entanglement in TQPTs.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
This paper investigates the topological unfolding of multi-body quantum entanglement, aiming to reveal the intrinsic structure and properties of entangled states. Utilizing topological methods, we mathematically analyze multi-body quantum entanglement and represent it as a series of topological structures. This approach offers a novel perspective on understanding complex entanglement phenomena, potentially leading to new insights in quantum information processing and fundamental physics. The core claim is to explore the topological unfolding of multi-body quantum entanglement, revealing the internal structure and properties of entanglement states. The core mechanism involves applying topological methods to mathematically analyze multi-body quantum entanglement, unfolding it into a series of topological structures. The novelty lies in the application of topology to multi-body quantum entanglement research, offering significant theoretical implications.