Quantum secret sharing scheme based on prime dimensional locally distinguishable states

In this paper, we first study the maximally commutative set in prime dimensional systems, which is a set of generalized Pauli matrices, and it can be used to detect the local discrimination of generalized Bell states. We give a simple characterization of prime dimensional maximally commutative sets, that is, a subset of a set of generalized Bell states, whose second subscript is a multiple of the first subscript. Furthermore, some sets of generalized Bell states which can be locally distinguishable by one-way local operation and classical communication (LOCC) are constructed by using the structural characteristics of prime dimensional maximally commutative sets. Based on these distinguishable generalized Bell states, we propose a (t, n)-threshold quantum secret sharing scheme. Compared with the existing quantum secret sharing scheme, it can be found that there are enough distinguishable states to encode classical information in our scheme, the dealer only needs to send entangled particles once to make the participants get their secret share, which makes the secret sharing process more efficient than the existing schemes. Finally, we prove that this protocol is secure under dishonest participant attack, interception-and-resend attack and entangle-and-measure attack.