  Quantum nonlocality offers a secure way to produce random numbers: their
unpredictability is intrinsic and can be certified just by observing the
statistic of the measurement outcomes, without assumptions on how they are
produced. To do this, entangled pairs are generated and measured to violate a
Bell inequality with the outcome statistics. However, after a projective
quantum measurement, entanglement is entirely destroyed and cannot be used
again. This fact poses an upper bound to the number of random numbers that can
be produced from each quantum state when projective measurements are employed.
Instead, by using weak measurements, some entanglement can be maintained and
reutilized, and a sequence of weak measurements can extract an unbounded amount
of randomness from a single state as predicted in Phys. Rev. A 95, 020102(R)
(2017). We study the feasibility of these weak measurements, analyze the
robustness to imperfections in the quantum state they are applied to, and then
test them using an optical setup based on polarization-entangled photon pairs.
We show that the weak measurements are realizable, but can improve the
performance of randomness generation only in close-to-ideal conditions.
