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Geometric Entanglement of Symmetric States and the Majorana Representation

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Theory of Quantum Computation, Communication, and Cryptography (TQC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6519))

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Abstract

Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum information tasks. This article builds upon the results of [New J. Phys. 12 (2010) 073025], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entanglement determined by numeric and analytic means. For this the Majorana representation, a generalization of the Bloch sphere representation, can be employed to represent symmetric n qubit states by n points on the surface of a unit sphere. Symmetries of this point distribution simplify the determination of the entanglement, and enable the study of quantum states in novel ways. Here it is shown that the duality relationship of Platonic solids has a counterpart in the Majorana representation, and that in general maximally entangled symmetric states neither correspond to anticoherent spin states nor to spherical designs. The usability of symmetric states as resources for measurement-based quantum computing is also discussed.

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Author information

Authors and Affiliations

  1. The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom

    Martin Aulbach

  2. Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom

    Martin Aulbach

  3. CNRS, LTCI, Telecom ParisTech, 23 Avenue d’Italie, 75013, Paris, France

    Damian Markham

  4. Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo, 113-0033, Japan

    Mio Murao

  5. Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo, 113-0033, Japan

    Mio Murao

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  1. Martin Aulbach
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  2. Damian Markham
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  3. Mio Murao
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of California, 93106-5110, Santa Barbara, CA, USA

    Wim van Dam

  2. School of Physics and Astronomy, University of Leeds, LS2 9JT, Leeds, UK

    Vivien M. Kendon

  3. Department of Physics and Astronomy, University College London, WC1E 6BT, London, UK

    Simone Severini

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Aulbach, M., Markham, D., Murao, M. (2011). Geometric Entanglement of Symmetric States and the Majorana Representation. In: van Dam, W., Kendon, V.M., Severini, S. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2010. Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18073-6_12

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  • DOI: https://doi.org/10.1007/978-3-642-18073-6_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-18072-9

  • Online ISBN: 978-3-642-18073-6

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