  Device-independent quantum key distribution (DIQKD) is the art of using
untrusted devices to establish secret keys over an untrusted channel. So far,
the real-world implementation of DIQKD remains a major challenge, as it
requires the demonstration of a loophole-free Bell test across two remote
locations with very high quality entanglement to ensure secure key exchange.
Here, we demonstrate for the first time the distribution of a secure key --
based on asymptotic security estimates -- in a fully device-independent way
between two users separated by 400 metres. The experiment is based on heralded
entanglement between two independently trapped single Rubidium 87 atoms. The
implementation of a robust DIQKD protocol indicates an expected secret key rate
of r=0.07 per entanglement generation event and r>0 with a probability error of
3%. Furthermore, we analyse the experiment's capability to distribute a secret
key with finite-size security against collective attacks.
