  In quantum cryptography, device-independent (DI) protocols can be certified
secure without requiring assumptions about the inner workings of the devices
used to perform the protocol. In order to display nonlocality, which is an
essential feature in DI protocols, the device must consist of at least two
separate components sharing entanglement. This raises a fundamental question:
how much entanglement is needed to run such DI protocols? We present a
two-device protocol for DI random number generation (DIRNG) which produces
approximately $n$ bits of randomness starting from $n$ pairs of arbitrarily
weakly entangled qubits. We also consider a variant of the protocol where $m$
singlet states are diluted into $n$ partially entangled states before
performing the first protocol, and show that the number $m$ of singlet states
need only scale sublinearly with the number $n$ of random bits produced.
Operationally, this leads to a DIRNG protocol between distant laboratories that
requires only a sublinear amount of quantum communication to prepare the
devices.
