Abstract
We introduce a new family of quantum secret sharing protocols with limited quantum resources which extends the protocols proposed by Markham and Sanders [14] and Broadbent, Chouha, and Tapp [2]. 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 \(k< \frac{79}{156}n\) 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 in [14] admit no threshold k when the secret is encoded on all the qubits and n > n 0.
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Javelle, J., Mhalla, M., Perdrix, S. (2013). New Protocols and Lower Bounds for Quantum Secret Sharing with Graph States. In: Iwama, K., Kawano, Y., Murao, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2012. Lecture Notes in Computer Science, vol 7582. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35656-8_1
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DOI: https://doi.org/10.1007/978-3-642-35656-8_1
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