Quantum key distribution between N partners: Optimal eavesdropping and Bell’s inequalities

Quantum secret-sharing protocols involving N partners are key distribution protocols in which Alice encodes her key into $N-1$ qubits, in such a way that all the other partners must cooperate in order to retrieve the key. On these protocols, several eavesdropping scenarios are possible: some partners may want to reconstruct the key without the help of the other ones, and consequently collaborate with an Eve that eavesdrops on the other partners’ channels. For each of these scenarios, we give the optimal individual attack that Eve can perform. A link with Bell’s inequality is demonstrated analytically for half of the scenarios, and is conjectured on the basis of numerical estimates for the other scenarios: the authorized partners have a higher information on the key than the unauthorized ones if and only if they can violate a Bell’s inequality.