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 [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 - n0.68 there exists a graph allowing a ((k, n)) protocol. On the other hand, we prove that for any k < 79/156n there is no graph G allowing a ((k, n)) protocol. As a consequence there exists n0 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 > n0. © Springer-Verlag Berlin Heidelberg 2013.