  Quantum systems are the ultimate touchstone for the production of random
sequences of numbers. Spatially spread entangled systems allow the generation
of identical random sequences in remote locations. The impossibility of
observing a quantum system, without disturbing it, ensures that the messages
encoded using these sequences cannot be eavesdropped. This is the basis of
Quantum Key Distribution. It is then of crucial importance knowing whether the
sequences generated in the practice by spatially spread entangled states are
truly random, or not. Yet, that knowledge is not immediate. One of the
obstacles is the very definition of randomness. Statistical randomness is
related with the frequency of occurrence of strings of data. On the other hand,
algorithmic randomness is related with the compressibility of the sequence,
what is given by Kolmogorov complexity. We analyze sequences generated by
entangled pairs of photons focusing on an estimation of their complexity.
