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The security of device-independent (DI) quantum key distribution (QKD)protocols relies on the violation of Bell inequalities. As such, their securitycan be established based on minimal assumptions about the devices, but theirimplementation necessarily requires the distribution of entangled states. In asetting with fully trusted devices, any entanglement-based protocol isessentially equivalent to a corresponding prepare-and-measure protocol. Thiscorrespondence, however, is not generally valid in the DI setting unless onemakes extra assumptions about the devices. Here we prove that a known tightlower bound on the min entropy in terms of the CHSH Bell correlator, which hasfeatured in a number of entanglement-based DI QKD security proofs, also holdsin a prepare-and-measure setting, subject only to the assumption that thesource is limited to a two-dimensional Hilbert space.

Secrecy in prepare-and-measure CHSH tests with a qubit bound