A quantum cloning bound and application to quantum key distribution

We introduce a quantum cloning bound which we apply to a straightforward and relatively direct security proof of the prepare-and-measure Bennett-Brassard 1984 (BB84) quantum key distribution (QKD) protocol against collective attacks. The approach we propose is able to handle the practical problem of source and detector alignment imprecisions in a simple way. Specifically, we derive a keyrate bound for a BB84 implementation in which Alice's source emits four given but arbitrary pure states, where the usual equivalence between prepare-and-measure and entanglement-based QKD no longer applies. Our result is similar to a keyrate derived by Mar{\o}y et. al. [Phys. Rev. A 82, 032337 (2010)] and generally an improvement over the keyrate derivable from the entropic uncertainty relation in situations where it applies. We also provide a stronger result for a source emitting arbitrary qubit states, and a further improved result if the detector is additionally assumed two dimensional.