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

TL;DR

This paper introduces a new family of quantum secret sharing protocols based on graph states, extending previous methods, and establishes bounds on the achievable thresholds for secret reconstruction.

Contribution

It presents a parametrized protocol using graph states for quantum secret sharing and derives bounds on the threshold values for different graph configurations.

Findings

Existence of graphs for thresholds greater than n - n^{0.68}.

Non-existence of graphs for thresholds less than 79n/156.

Protocols by Markham and Sanders cannot achieve certain thresholds for large n.

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.