Randomness and Bell inequalities

Randomness has always been a matter of trust. For instance, we trust that nobody is able to predict the outcome of a die from the position of the hand at the moment of rolling it. However, we would regret gambling when told that our die has been previously intervened by a group of experts on robotics. Indeed, physicists face a similar problem when addressing the question of randomness in Nature. They make use of models and theories that they trust and that in some cases conclude that a process is random. But they can never be sure that Nature is not playing robotics somewhere behind and yet nobody noticed.

Bell inequalities to certify quantum dice. (source: www.CGPrey.com)

A seminal case of a random theory is Quantum Mechanics. According to this theory, one can find random phenomena in almost every microscopic process, to the extent that fully predictable events are a rare exception at the atomic scale. Hence, it seems quite straightforward to build 'quantum dice' which are totally unpredictable. However this sounds appealing, the situation is not much different: Randomness is still a matter of trust. If one offers us quantum dice with a working mechanism based on the decay of radioactive atom –one of the random phenomena in quantum mechanics–, do we need to trust the vendor about the internal mechanism? How can we tell if the mechanism gets degraded with time? Or even worse, why should one trust quantum mechanics? Indeed, it is not difficult to imagine, that in the future an improved theory replaces quantum mechanics. Such theory may be able to predict perfectly all these phenomena that seemed random in the 'old quantum theory' and all quantum dice would end up in the garbage.

In recent years, new venues to approach the problem of randomness have been open with the use of Bell's theorem. The dream is to design an experiment whose outcomes can be certified to be random, without having to trust the vendor of the machine or the capabilities of your adversary. Contrary to all the previous examples, such randomness would not be a matter of trust. Surprisingly enough, a closely related experiment can be performed by using Bell inequalities and was first realized in a laboratory employing the CHSH inequality. The idea is that two distant observers measure on their share of a quantum system. They make use of an initial source of randomness: a reduced number of bits that are assumed to be random. If they violate the CHSH inequality then one is certain that many new random bits have been generated in the process. In other words Randomness can be amplified.

This discovery provoked an explosion of interests on the topic and more sophisticated devices employing other Bell inequalities were put forward. Using the Chained and the Mermin Bell inequalities, it has been shown that such randomness amplification can be made infinite: it produces infinite random numbers from an arbitrarily weak initial source. More importantly, it has been shown that one does not even have to trust on quantum mechanics. These devices produce random numbers even if physicists are mistaken about quantum theory. We can now safely say that Nature has little weight to load the dice.

Rodrigo Gallego

Dahlem Center for Complex Quantum Systems
Freie Universität Berlin
14195 Berlin

published online on 4 Feb 2014