Session W30: Quantum Entanglement

Entanglement, Fluctuations, and Quantum Critical Points

We show that bipartite fluctuations F can be considered an entanglement measure. We further demonstrate that the concept of bipartite fluctuations F provides a very efficient tool to detect quantum phase transitions in strongly correlated systems. We investigate paradigmatic examples for both quantum spins and bosons in one and two dimensions. As compared to the von Neumann entanglement entropy, we observe that F allows to find quantum critical points with a much better accuracy in one dimension. We further demonstrate that F can be successfully applied to the detection of quantum criticality in higher dimensions with no prior knowledge of the universality class of the transition. Promising approaches to experimentally access fluctuations are discussed for quantum antiferromagnets and cold gases.   

A statistical portrait of the entanglement decay of two-qubit memories

We present a novel approach to the study of entanglement decay, which focuses on collective properties. As an example, we investigate the entanglement decay of a two-qubit memory, produced by local identical reservoirs acting on the qubits, for three experimentally and theoretically relevant cases: depolarizing, dephasing and amplitude-damping channels. We study the probability distributions of disentanglement times, a quantity independent of the measure used to quantify entanglement, and the time-dependent probability distribution of concurrence. Uniformly distributed pure states are assumed for the two-qubit system. The calculation of these probability distributions gives a clearer insight on how different decoherence channels affect the entanglement initially contained in the set of two-qubit pure states. The entanglement evolution of mixed states, under the Hilbert-Schmidt metric, is also considered.   

Quantum Mutual Information Capacity for High Dimensional Entangled States

High dimensional Hilbert spaces used for quantum communication channels offer the possibility of large data transmission capabilities and improved security. We propose a method of characterizing the channel capacity of an entangled photonic state in high dimensional position and momentum bases. We use this method to measure the channel capacity of a parametric downconversion state, achieving a channel capacity over 7 bits/photon in either the position or momentum basis, by measuring in up to 576 dimensions per detector. The channel strongly violated an entropic separability bound, indicating the performance cannot be replicated classically.   

Entangling Qubits in a One-Dimensional Harmonic Oscillator

We present a method for generating entanglement between qubits associated with a pair of particles interacting in a one-dimensional harmonic potential. By considering the effect of the interaction on the energy spectrum of the system, we show that, under certain approximations, a ``power-of-SWAP" operation is performed on the initial two-qubit quantum state without requiring any time-dependent control. Initialization errors and deviations from our approximation are shown to have a negligible effect on the final state. Using a GPU-accelerated iteration scheme to find numerical solutions to the two-particle time-dependent Schr\"{o}dinger equation, we demonstrate that it is possible to generate maximally entangled Bell states between the two qubits with high fidelity for a range of possible interaction potentials.   

Entanglement entropy for arbitrary quantum lattice models from quantum Monte Carlo

We present a general scheme to numerically calculate the Renyi entropy for the reduced density matrix of a subsystem in a quantum lattice model at finite and (physically) zero temperature. This scheme is based on an extended-ensemble formulation of quantum Monte Carlo, which can be applied in principle to any quantum Monte Carlo algorithm. It improves on the existing approach of R. G. Melko et al., Phys. Rev. B 82, 100409(R) (2010) and of M. B. Hastings et al., Phys. Rev. Lett. 104, 157201 (2010) in that it allows to probe the ground-state properties of lattice models regardless of their symmetry - as long as they admit an efficient quantum Monte Carlo algorithm. We test the entanglement entropy scaling of fundamental quantum spin models, showing e.g. that the two-dimensional XX model, describing lattice hardcore bosons, exhibits an area law despite lacking an intrinsic length scale for the decay of correlations.   

Different Measures of Entanglement in Spin Chains

Different measures of entanglement in spin chains are considered. Main example is VBS state, it is important because of measurement based quantum computation. Entanglement spectrum and negativity are considered in the lecture. These measures are calculated analytically in one dimension. In 2D we have only estimates. Lecture follows the papers: http://arxiv.org/abs/1109.4971 and http://arxiv.org/abs/1110.3300 \\[4pt] [1] Heng Fan, Vladimir Korepin, Vwani Roychowdhury, PRL, {\bf vol 93}, (2004), 227203   

R\'enyi entropy of $d$-wave Bose metal phases on multi-leg ladders

In recent years, much progress has been made toward understanding 2D Bose metal-type phases by accessing them through a series of controlled quasi-1D ladder studies [1]. Crucially, such quasi-1D descendants of these exotic phases are expected to have a number of gapless Luttinger modes, $c$, that grows with the width of the ladder. Therefore, characterizing scaling of the entanglement entropy has become an essential tool for establishing the existence of these phases, as it provides a direct measure of $c$. With density-matrix renormalization group (DMRG) methods, it is easy to calculate the entanglement entropy but the results converge prohibitively slowly as the ladder becomes wider. Here, we present results where we have calculated, using Variational Monte Carlo (VMC), the R\'enyi entropy $S_2$ for Gutzwiller-projected $d$-wave Bose metal (DBM) [2] trial wave functions on ladders, following the method employed in [3]. We compare with DMRG results (where they are available), and comment on what our findings mean for the ability of our trial wave functions to faithfully represent the DBM phase. \\[4pt] [1] D. N. Sheng et. al., PRB {\bf 78}, 054520 (2008).\\[0pt] [2] O. I. Motrunich and M. P. A. Fisher, PRB {\bf 75}, 235116 (2007).\\[0pt] [3] Y. Zhang et. al., PRL {\bf 107}, 067202 (2011).   

R\'{e}nyi Entanglement Entropies and the Entanglement Spectrum

We describe a simple method for computing the full entanglement spectrum of any finite density matrix from the R\'{e}nyi entropies of integer order. This has important implications for non-interacting fermionic systems where the R\'{e}nyi entropies are directly related to the cumulants of charge number fluctuations, and for quantum Monte Carlo simulations where it is now becoming possible to compute the R\'{e}nyi entropies but not the von Neumann entropy or the full entanglement spectrum.   

Entanglement Entropy of Fermi Liquids via Multi-dimensional Bosonization

The logarithmic violations of the area law, i.e. an ``area law'' with logarithmic correction of the form $S \sim L^{d-1} \log L$, for entanglement entropy are found in both 1D gapless system and for high dimensional free fermions. The purpose of this work is to show that both violations are of the same origin, and in the presence of Fermi liquid interactions such behavior persists for 2D fermion systems. In this paper we first consider the entanglement entropy of a toy model, namely a set of decoupled 1D chains of free spinless fermions, to relate both violations in an intuitive way. We then use multi-dimensional bosonization to re-derive the formula by Gioev and Klich [Phys. Rev. Lett. 96, 100503 (2006)] for free fermions through a low-energy effective Hamiltonian, and explicitly show the logarithmic corrections to the area law in both cases share the same origin: the discontinuity at the Fermi surface (points). In the presence of Fermi liquid (forward scattering) interactions, the bosonized theory remains quadratic in terms of the original local degrees of freedom, and after regularizing the theory with a mass term we are able to calculate the entanglement entropy perturbatively up to second order in powers of the coupling parameter for a special geometry via the replica trick.   

Multipartite Entanglement classes via Negativity Fonts

The number and types of K-way negativity fonts in canonical form of an N-qubit state depends on the nature and amount of quantum coherences in the state. Non zero determinants of negativity fonts, characterizing a given state, are easily written down and reflect the entanglement microstructure of the superposition. A classification criterion for multipartite entangled states, based on negativity fonts in canonical state and decomposition of global partial transpose in terms of K-way partially transposed operators, is proposed. Inequivalent sub-classes are labelled by N-qubit local unitary invariants. A complete classification of four qubit states is obtained. The number of major families for N$>$3 is found to be $2^N-2N$. Classification of four qubit states indicates that a small number of relevant polynomial invariants is enough to classify N-qubit states.   

Entanglement Spectrum Classification of Disordered Class AII Symplectic Systems

Of the available classes of random matrices which have been shown to contain topologically non-trivial properties\footnote{A.~P.~Schnyder, S.~Ryu, A.~Furusaki, and A~.W.~W.~Ludwig, \emph{Phys. Rev. B} \textbf{55}, 195125 (2008).}, one of the most intriguing is class AII, which is characterizes a system that possesses time-reversal symmetry. This class of random matrices has been the subject of significant attention as it encompasses Z$_2$ topological systems of which the quantum spin Hall (QSH) state is a member~\footnote{C.~L.~Kane and E.~J.~Mele, \emph{Phys. Rev. Lett.} \textbf{95}, 146802 (2005).}. We calculate the entanglement spectrum for disordered class AII symplectic systems in two-dimensions as a function of disorder strength, chemical potential, and bulk inversion asymmetry. We show that there is a one to one correspondence between the full system Hamiltonian and that of the entanglement spectrum not only in terms of level statistics but also in terms of the scaling of the inverse participation ratios. We also use the properties of the entanglement spectrum to illustrate the nature of the symplectic metal phase which appears when inversion symmetry is broken.   

Experimental entanglement estimation for a general unknown input state of a multiqubit system

Measurement of entanglement remains an important problem for quantum information. We present the design and simulation of an experimental method for entanglement estimation for a general, unknown, state of a multiqubit system. The state can be in pure or mixed, and it need not be ``close'' to any particular state. Our method, based on dynamic learning, does not require prior state reconstruction or lengthy optimization. Results for three-qubit systems compare favorably with known entanglement measures. The method is then extended to four- and five-qubit systems, with relative ease. As the size of the system grows the amount of training necessary diminishes, raising hopes for applicability to large computational systems.   

Entanglement Spectrum of the Kugel-Khomskii Model

We study the entanglement spectrum of the Kugel-Khomskii model in one dimension. The Kugel-Khomskii Hamiltonian describes transition metal oxides with orbital degeneracy, and is rich with both gapless and gapped phases with interesting symmetries. The entanglement spectrum reveals much more information than the commonly studied entanglement entropy. In this work, we investigate the entanglement spectrum for different phases and different partitions. We also make comparisons with previous field theoretic and numerical studies.   

Entanglement Spectrum In Topological Phasess

I will review the information that entanglement spectra give for a wide range of systems in condensed matter physics, such as fractional quantum hall effect, quantum spin chains, topological insulators, and disordered systems. I will also show how the entanglement spectrum is a unique tool to examine previously unknown many-body wavefunctions such as the ground-states of Fractional Chern Insulators (the results are based on a series of works performed in collaboration with N. Regnault, M. Hermanns, B. Estienne, Yangle Wu, Aris Alexandadinata, R. Thomale, A Sterdyniak, Z. Papic, T.L. Hughes, E. Prodan, D.P. Arovas, P. Bonderson)   

Topological Order in Three Dimensions and Entanglement Entropy of Gapped Phases

In this talk, I will present two very general, yet easy to understand results in the entanglement entropy of the ground states corresponding to the gapped phases of matter. In particular, I will focus on the following two results: 1) In contrast to the familiar result in two dimensions, a size independent constant contribution to the entanglement entropy can appear for non-topological phases in any odd spatial dimension. 2) The ``topological entanglement entropy'' corresponding to discrete gauge theories in any given spatial dimension $D$ (and in particular, $D = 3$) has an interesting dependence on the Betti numbers of the boundary manifold defined by the entanglement cut.