Maximal secret reconstruction, teleportation and Bell’s inequality

Abstract

A tripartite state is said to be a potential resource for secret sharing if in addition to being useful for the secret reconstruction (Singh and Chakrabarty in: Phys Rev A 109(3):032406, 2024), it imposes restrictions on the teleportation fidelity of the bipartite channels associated with three-qubit states (dealer–reconstructor and dealer–assistant channels). It is important to ask the question: for a given class of states satisfying some constraint, which secret sharing resources will have the maximum possible reconstruction fidelity? Here, we address this question for a pure three-qubit GHZ class of states (sometimes referred as Acin states) (Antonio Acín et al. in: J Phys A Math Gen 34(35):6725, 2001; Acín et al. in: Phys Rev Lett 87(4):040401, 2001). We are able to characterize the set of states with maximum possible reconstruction fidelity (called as maximal secret reconstructible state [MSR]). Here, the constraint in characterizing the states is a fixed value of the maximum of the teleportation fidelity of both the bipartite (dealer–receivers) channels. In that spirit our result paves the way in setting the practical information transfer limit in a possible resource theoretic extension of secret sharing. Similarly for a value giving the maximum of Bell-CHSH value of both bipartite channels (dealer–reconstructor and dealer–assistant), we are able to find the maximum achievable reconstruction fidelity. Interestingly, we find that all secret shareable states satisfy Bell’s inequality in both the channels (dealer–reconstructor and dealer–assistant partitions). This brings out a new mutual exclusivity between secret shareable state and Bell’s inequality violation.