New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States

Abstract : We introduce a new family of quantum secret sharing protocols with limited quantum resources which extends the protocols proposed by Markham and Sanders and by Broadbent, Chouha, and Tapp. Parametrized by a graph G and a subset of its vertices A, the protocol consists in: (i) encoding the quantum secret into the corresponding graph state by acting on the qubits in A; (ii) use a classical encoding to ensure the existence of a threshold. These new protocols realize ((k,n)) quantum secret sharing i.e., any set of at least k players among n can reconstruct the quantum secret, whereas any set of less than k players has no information about the secret. In the particular case where the secret is encoded on all the qubits, we explore the values of k for which there exists a graph such that the corresponding protocol realizes a ((k,n)) secret sharing. We show that for any threshold k> n-n^{0.68} there exists a graph allowing a ((k,n)) protocol. On the other hand, we prove that for any k< 79n/156 there is no graph G allowing a ((k,n)) protocol. As a consequence there exists n_0 such that the protocols introduced by Markham and Sanders admit no threshold k when the secret is encoded on all the qubits and n>n_0.
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https://hal.archives-ouvertes.fr/hal-00934531
Contributor : Mehdi Mhalla <>
Submitted on : Wednesday, January 22, 2014 - 11:19:54 AM
Last modification on : Friday, October 25, 2019 - 2:00:50 AM

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Jérôme Javelle, Mehdi Mhalla, Simon Perdrix. New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States. TQC 2012 - 7th Conference on Theory of Quantum Computation, Communication, and Cryptography, May 2012, Tokyo, Japan. pp.1-12, ⟨10.1007/978-3-642-35656-8_1⟩. ⟨hal-00934531⟩

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