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arXiv:1612.02437

Multi-partite entanglement

Michael Walter, Gross, David, Jens Eisert

Published Dec 7, 2016• 19 citations
arXivHTMLPDF

Abstract

We give an introduction to the theory of multi-partite entanglement. We begin by describing the "coordinate system" of the field: Are we dealing with pure or mixed states, with single or multiple copies, what notion of "locality" is being used, do we aim to classify states according to their "type of entanglement" or to quantify it? Building on the general theory of multi-partite entanglement - to the extent that it has been achieved - we turn to explaining important classes of multi-partite entangled states, including matrix product states, stabilizer and graph states, bosonic and fermionic Gaussian states, addressing applications in condensed matter theory. We end with a brief discussion of various applications that rely on multi-partite entangled states: quantum networks, measurement-based quantum computing, non-locality, and quantum metrology.

DOI
10.48550/arxiv.1612.02437

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We give an introduction to the theory of multi-partite entanglement. We begin by describing the "coordinate system" of the field: Are we dealing with pure or mixed states, with single or multiple copies, what notion of "locality" is being used, do we aim to classify states according to their "type of entanglement" or to quantify it? Building on the general theory of multi-partite entanglement - to the extent that it has been achieved - we turn to explaining important classes of multi-partite entangled states, including matrix product states, stabilizer and graph states, bosonic and fermionic Gaussian states, addressing applications in condensed matter theory. We end with a brief discussion of various applications that rely on multi-partite entangled states: quantum networks, measurement-based quantum computing, non-locality, and quantum metrology.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
We review the theory of multi-particle entanglement. In this book chapter we aim at briefly ``setting the coordinates'' and guiding through the extensive literature in this field. Our coordinate system chosen for this chapter has the axes labeled pure and mixed states on the one hand, entanglement in single specimens and the asymptotic setting on the other hand. We very briefly mention ways to detect multi-particle entanglement, and introduce the concepts of stabilizer and graph states.
A concise introduction to quantum entanglement in multipartite systems is presented. We review entanglement of pure quantum states of three--partite systems analyzing the classes of GHZ and W states and discussing the monogamy relations. Special attention is paid to equivalence with respect to local unitaries and stochastic local operations, invariants along these orbits, momentum map and spectra of partial traces. We discuss absolutely maximally entangled states and their relation to quantum error correction codes. An important case of a large number of parties is also analysed and entanglement in spin systems is briefly reviewed.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.