Entropy Bounds for Multiparty Device-Independent Cryptography

Abstract

Multiparty quantum cryptography based on distributed entanglement will find its natural application in the upcoming quantum networks. The security of many multipartite device-independent (DI) protocols, such as DI conference-key agreement, relies on bounding the von Neumann entropy of the parties’ outcomes conditioned on the eavesdropper’s information, given the violation of a multipartite Bell inequality. We consider three parties testing the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality and certify the privacy of their outcomes by bounding the conditional entropy of a single party’s outcome and the joint conditional entropy of two parties’ outcomes. From the former bound, we show that genuine multipartite entanglement is necessary to certify the privacy of a party’s outcome, while the latter significantly improves previous results. We obtain the entropy bounds thanks to two general results of independent interest. The first one drastically simplifies the quantum setup of an N-partite Bell scenario. The second one provides an upper bound on the violation of the MABK inequality by an arbitrary N-qubit state, as a function of the state’s parameters.