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International Journal of Modern Physics B

ENTANGLEMENT AND ITS MULTIPARTITE EXTENSIONS

Andreas Osterloh

Published Oct 10, 2012• 9 citations
Publisher

Abstract

The aspects of many particle systems, as far as their entanglement is concerned, is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.

DOI
10.1142/s0217979213450185

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The aspects of many particle systems, as far as their entanglement is concerned, is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.
This thesis poses a selection of recent research of the author in a common context. It starts with a selected review on research concerning the role entanglement might play at quantum phase transitions and introduces measures for entanglement used for this analysis. A selection of results from this research is given and proposed as evidence for the relevance of multipartite entanglement in this context. A constructive method for an SLOCC classification and quantification of multipartite qubit entanglement is outlined and results for convex roof extensions of the resulting measures are briefly discussed on a specific example. At the end, a transformation of antilinear expectation values into linear expectation values is presented which admits an expression of the aforementioned measures of genuine multipartite entanglement in terms of spin correlation function, hence making them experimentally accessible.
This paper addresses the following main question: Do we have a theoretical understanding of entanglement applicable to a full variety of physical settings? It is clear that not only the assumption of distinguishability, but also the few-subsystem scenario, are too narrow to embrace all possible physical settings. In particular, the need to go beyond the traditional subsystem-based framework becomes manifest when one tries to apply the conventional concept of entanglement to the physics of matter, since the constituents of a quantum many-body system are indistinguishable particles. We shall discuss here a notion of generalized entanglement, which can be applied to any operator language (fermions, bosons, spins, etc.) used to describe a physical system and which includes the conventional entanglement settings introduced to date in a unified fashion. This is realized by noticing that entanglement is an observer-dependent concept, whose properties are determined by the expectations of a distinguished set of observables without reference to a preferred subsystem decomposition, i.e., it depends on the physically relevant point of view. This viewpoint depends in turn upon the relationship between different sets of observables that determine our ability to control the system of interest. Indeed, the extent to which entanglement is present depends on the observables used to measure a system and describe its states. This represents a most conspicuous advantage as will be highlighted by the condensed-matter application we will discuss.
The study and characterisation of quantum entanglement represents an extended effort within the field of quantum information. This thesis addresses two problems. The first regards entanglement in open systems, where it’s more difficult to define a dichotomic (two outcome) observable due to population loss. The second problem concerns genuine multipartite entanglement, that is entanglement between many particles in states that cannot be separable with respect to any possible bipartition of a system.\nThe first chapter consists of an introduction to the basic elements of quantum mechanics and quantum information theory. We briefly describe a few historical facts about quantum mechanics and the study of entanglement, to help put things into perspective.\nThe second chapter is a detailed discussion of the neutral kaon system. These particles are produced in pairs and are described as a particle-antiparticle system. This system displays some unique properties like the violation of CP-symmetry, and shares some characteristics with other mesons, such as neutral particle oscillation and decay. This makes formulating Bell-type inequalities for them more difficult, but also provides the opportunity for new tests.\nThe third chapter continues with the presentation of an already existent effective formalism for performing Bell tests on neutral kaons. The essence of the formalism is a switch to the Heisenberg picture, transferring the time evolution and dependence on measurement directions to a so-called effective operator. This has many advantages, some of the most important being a proper normalization during population loss (due to decay), an easy application to\nthe case of multiple particles and the fact that it allows one to observe entanglement within the system for a relatively long amount of time. The final part presents a way to simulate neutral kaons with atomic systems, namely with Ytterbium isotopes and discuss what are the differences between these atomic systems and neutral kaons.\nThe second part of the thesis treats the problem of proving genuine multipartite entanglement from separable two-body marginals. For this, an introduction to semidefinite programming is necessary due to the fact that the problem under study can be formulated in terms of this method. We will briefly discuss linear and semidefinite programming as well as the concept of duality and provide some basic examples.\nThe formal description of this problem and previous results are provided in the fifth chapter. Our result shows that, through a cyclic iteration between two SDPs, one obtains states with the desired properties. Using this method we managed to go up to six qubit states for various configurations. From this we can construct higher-dimensional states having the same properties.
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two bosons or fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated as absence of nonlocal correlation among two different sets of modes. While the first approach applies to pure states only, the second one is more general and characterizes generic entangled states. In the following, we shall show that the mode-based approach indeed contains the particle-based one.
We present a method to construct entanglement measures for pure states of multipartite qubit systems. The key element of our approach is an antilinear operator that we call comb in reference to the hairy-ball theorem. For qubits (i.e. spin 1/2) the combs are automatically invariant under SL (2, ℂ). This implies that the filters obtained from the combs are entanglement monotones by construction. We give alternative formulae for the concurrence and the 3-tangle as expectation values of certain antilinear operators. As an application we discuss inequivalent types of genuine four-, five- and six-qubit entanglement.