  Quantum theory allows for randomness generation in a device-independent
setting, where no detailed description of the experimental device is required.
Here we derive a general upper bound on the amount of randomness that can be
generated in such a setting. Our bound applies to any black-box scenario, thus
covering a wide range of scenarios from partially characterised to completely
uncharacterised devices. Specifically, we prove that the number of random bits
that can be generated is limited by the number of different input states that
enter the measurement device. We show explicitly that our bound is tight in the
simplest case. More generally, our work indicates that the prospects of
generating a large amount of randomness by using high-dimensional (or even
continuous variable) systems will be extremely challenging in practice.
