Security and entanglement in differential-phase-shift quantum key distribution

The differential-phase-shift quantum key distribution protocol is formalised as a prepare-and-measure scheme and translated into an equivalent entanglement-based protocol. A necessary condition for security is that Bob's measurement can detect the entanglement of the distributed state in the entanglement-based translation, which implies that his measurement is described by non-commuting POVM elements. This condition is shown to be met.