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arXiv:quant-ph/0010052 · Physical Review A

Potential multiparticle entanglement measure

Alexander K.C. Wong, Nelson L Christensen

Published Mar 13, 2001• 297 citations
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Abstract

In this Brief Report we discuss entanglement of multiparticle quantum systems. We propose a potential measure of a type of entanglement of pure states of $n$ qubits, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}.$ For a system of two qubits the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is equal to the square of the concurrence, and for systems of three qubits it is equal to the ``residual entanglement.'' We show that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is also equal to a generalization of the concurrence squared for even $n,$ and use this fact to prove that the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is an entanglement monotone. However, the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ is undefined for odd $n>3.$ Finally, we propose a measure related to the $n\ensuremath{-}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}$ for mixed-state systems of $n$ qubits, and find an analytical formula for this measure for even $n.$

DOI
10.1103/physreva.63.044301

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Abstract Entanglement is a key ingredient for quantum technologies and a fundamental signature of quantumness in a broad range of phenomena encompassing many-body physics, thermodynamics, cosmology and life sciences. For arbitrary multiparticle systems, entanglement quantification typically involves nontrivial optimisation problems, and it may require demanding tomographical techniques. Here, we develop an experimentally feasible approach to the evaluation of geometric measures of multiparticle entanglement. Our framework provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, or genuine multiparticle entanglement of any general state. For global and partial entanglement, useful bounds are obtained with minimum effort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N are required. We demonstrate the power of our approach to estimate and quantify different types of multiparticle entanglement in a variety of N-qubit states useful for quantum information processing and recently engineered in laboratories with quantum optics and trapped ion setups.
We investigate parametrized multipartite entanglement measures from the perspective of kkk nonseparability in this paper. We present two types of entanglement measures in nnn-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all kkk-nonseparable states in arbitrary nnn-partite systems. Rigorous proofs show that the proposed kkk-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all kkk-separable states, and being strictly greater than zero for all kkk-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition (k=2)(k=2)(k=2). The lower bounds of the two classes kkk-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.
We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multipartite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space, and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states.
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Multiparticle entanglement is a useful resource in quantum information processing. It is involved in some quantum key distribution protocols, quantum metrology and many other physical applications and phenomena and can be experimentally observed in various quantum systems. Having said this, its classification, detection and especially its quantification is quite challenging. To this day there exists no general mixed state measure for genuine multiparticle entanglement, which can be computed and analytically treated at the same time.\n\nIn this thesis the analytical characterisation of genuine multiparticle entanglement in quantum systems using the computable genuine multiparticle negativity as entanglement measure is provided. Furthermore, the notion of stabiliser states, which are families of symmetric genuine multiparticle entangled states, is generalised and a useful method to exploit local symmetries to speed up the computation of the investigated entanglement measure is provided.