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Quantum Secret Sharing with Graph States

  • Sylvain Gravier
  • Jérôme Javelle
  • Mehdi Mhalla
  • Simon Perdrix
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7721)

Abstract

We study the graph-state-based quantum secret sharing protocols [24,17] which are not only very promising in terms of physical implementation, but also resource efficient since every player’s share is composed of a single qubit. The threshold of a graph-state-based protocol admits a lower bound: for any graph of order n, the threshold of the corresponding n-player protocol is at least 0.506n. Regarding the upper bound, lexicographic product of the C 5 graph (cycle of size 5) are known to provide n-player protocols which threshold is n − n 0.68. Using Paley graphs we improve this bound to n − n 0.71. Moreover, using probabilistic methods, we prove the existence of graphs which associated threshold is at most 0.811n.Albeit non-constructive, probabilistic methods permit to prove that a random graph G of order n has a threshold at most 0.811n with high probability. However, verifying that the threshold of a given graph is acually smaller than 0.811n is hard since we prove that the corresponding decision problem is NP-Complete.These results are mainly based on the graphical characterization of the graph-state-based secret sharing properties, in particular we point out strong connections with domination with parity constraints.

Keywords

Quantum Information Graph Theory Quantum Cryptography NP-Completeness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sylvain Gravier
    • 1
    • 2
  • Jérôme Javelle
    • 3
  • Mehdi Mhalla
    • 1
    • 3
  • Simon Perdrix
    • 1
    • 3
  1. 1.CNRSFrance
  2. 2.Institut FourierUniversity of GrenobleFrance
  3. 3.LIGUniversity of GrenobleFrance

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