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arXiv:1904.07782 · New Journal of Physics

Measuring von Neumann entanglement entropies without wave functions

T Mendes-Santos, G Giudici, Rosario Fazio, Marcello Dalmonte

Published Jan 1, 2020• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/ab6875

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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.
In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.
Measuring entanglement entropy in interacting, multipartite systems remains a significant experimental challenge. We address this challenge by developing a protocol to measure von Neumann entropy (VNE) and mutual information in quantum transport systems with both many-body interactions and multiple subsystems. Our analysis indicates that the vital connection between VNE and two-point correlation functions persists under these realistic conditions. The measurement is shown to be feasible for systems with boundary interactions and, critically, for bulk-interacting systems subject to a quantum quench of their internal couplings. Our work provides a pathway to experimentally quantify entanglement in complex interacting systems and establishes mutual information as an experimentally accessible indicator for system-environment entanglement.
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.
Recent advances in the field of strongly correlated electron systems allow to access the entanglement properties of interacting fermionic models, by means of Monte Carlo simulations. We briefly review the techniques used in this context to determine the entanglement entropies and correlations of the entanglement Hamiltonian. We further apply these methods to compute the spin two-point function of entanglement Hamiltonian for a stripe embedded into a correlated topological insulator. Further we discuss a recent method that allows an unbiased, numerically exact, direct determination of the entanglement Hamiltonian by means of auxiliary field quantum Monte Carlo simulations.
Entanglement entropy encodes important features of strongly interacting quantum many-body systems and gauge theories, but its analytical study is still limited to systems with high level of symmetry. This motivates the search for efficient techniques to investigate this quantity numerically, through Monte Carlo calculations on the lattice. In this contribution, we discuss the computation of the entropic c-function by means of an algorithm based on Jarzynski's equality, which is an exact theorem from non-equilibrium statistical mechanics. After presenting benchmark results for the Ising model in two dimensions, where our algorithm successfully reproduces the analytical predictions from conformal field theory, we discuss its generalization to the three-dimensional Ising model, for which we were able to extract universal terms beyond the area law. Finally we point out some future generalizations of this calculation.
We construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to local Hamiltonians. Our focus is on entanglement entropies of subsystems that make up approximately half of the full system. The upper bound on the von Neumann entanglement entropy is half the sum of the thermal entropies of two fictitious systems at the same temperature as one another, with an additional area-law contribution in some systems. The effective temperature is chosen such that the sum of the thermal energies of the two fictitious systems matches the constraint on the energy of the state in the original problem; at subextensive energies, this temperature decreases with increasing system size. Our upper bounds on Rényi entanglement entropies take an analogous form. These inequalities show that, in systems that are frustration-free (FF) and whose zero-temperature subsystem thermal entropies scale with surface areas (rather than subsystem volumes), ground states are area-law entangled. For physical models of quantum matter, which have well-defined specific heat capacities (and are not necessarily FF), our bounds provide a way to convert this thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies. We also show that our general upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in systems with reflection symmetry and above a subextensive energy threshold. A key application of our results is to constrain the entanglement of quantum states relevant to simulations of zero-temperature response functions, which have energies of order unity. We prove that, in wide varieties of FF systems in two dimensions and higher, the contributions of excitations to von Neumann entanglement entropy are subleading compared with the contribution from the ground state.