Quantum Secret Sharing with Graph States

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.
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Sylvain Gravier, Jérôme Javelle, Mehdi Mhalla, Simon Perdrix. Quantum Secret Sharing with Graph States. MEMICS 2012 - International Doctoral Workshop on Mathematical and Engineering Methods in Computer Science, Oct 2012, Znojmo, Czech Republic. pp.15-31, ⟨10.1007/978-3-642-36046-6_3⟩. ⟨hal-00933722⟩

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