  According to the entropy accumulation theorem, proving the unconditional
security of a device-independent quantum key distribution protocol reduces to
deriving tradeoff functions, i.e., bounds on the single-round von Neumann
entropy of the raw key as a function of Bell linear functionals, conditioned on
an eavesdropper's quantum side information. In this work, we describe how the
conditional entropy can be bounded in the 2-input/2-output setting, where the
analysis can be reduced to qubit systems, by combining entropy bounds for
variants of the well-known BB84 protocol with quantum constraints on qubit
operators on the bipartite system shared by Alice and Bob. The approach gives
analytic bounds on the entropy, or semi-analytic ones in reasonable computation
time, which are typically close to optimal. We illustrate the approach on a
variant of the device-independent CHSH QKD protocol where both bases are used
to generate the key as well as on a more refined analysis of the original
single-basis variant with respect to losses. We obtain in particular a
detection efficiency threshold slightly below 80.26%, within reach of current
experimental capabilities.
