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Entanglement of the Antisymmetric State

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Abstract

We analyse the entanglement of the antisymmetric state in dimension d × d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by \({O(\frac{1}{d})}\). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state.

Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results.

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Authors and Affiliations

  1. Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093, Zurich, Switzerland

    Matthias Christandl

  2. Institute for Quantum Information, California Institute of Technology, Pasadena, CA, 91125, USA

    Norbert Schuch

  3. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748, Garching, Germany

    Norbert Schuch

  4. Department of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Andreas Winter

  5. Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117542, Singapore, Singapore

    Andreas Winter

Authors
  1. Matthias Christandl
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  2. Norbert Schuch
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  3. Andreas Winter
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Correspondence to Matthias Christandl.

Additional information

Communicated by M. B. Ruskai

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Christandl, M., Schuch, N. & Winter, A. Entanglement of the Antisymmetric State. Commun. Math. Phys. 311, 397–422 (2012). https://doi.org/10.1007/s00220-012-1446-7

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  • DOI: https://doi.org/10.1007/s00220-012-1446-7

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