  Device-independent quantum key distribution (DIQKD) aims to mitigate
adversarial exploitation of imperfections in quantum devices, by providing an
approach for secret key distillation with modest security assumptions.
Advantage distillation, a two-way communication procedure in error correction,
has proven effective in raising noise tolerances in both device-dependent and
device-independent QKD. Previously, device-independent security proofs against
IID collective attacks were developed for an advantage distillation protocol
known as the repetition-code protocol, based on security conditions involving
the fidelity between some states in the protocol. However, there exists a gap
between the sufficient and necessary security conditions, which hinders the
calculation of tight noise-tolerance bounds based on the fidelity. We close
this gap by presenting an alternative proof structure that replaces the
fidelity with the quantum Chernoff divergence, a distinguishability measure
that arises in symmetric hypothesis testing. Working in the IID collective
attacks model, we derive matching sufficient and necessary conditions for the
repetition-code protocol to be secure (up to a natural conjecture regarding the
latter case) in terms of the quantum Chernoff divergence, hence indicating that
this serves as the relevant quantity of interest for this protocol.
Furthermore, using this security condition we obtain some improvements over
previous results on the noise tolerance thresholds for DIQKD. Our results
provide insight into a fundamental question in quantum information theory
regarding the circumstances under which DIQKD is possible.
