Three hats, which can be black or white, are placed on three people. Each one does not know what color her hat is, but she does see that of the others. At the beginning, they are told that there is at least one white hat. If you ask them, does anyone know what color their hat is? They will say no. If you ask again, they will say no again. But the third time you ask them, they will all know. Well, what color are their hats?
This famous riddle serves to illustrate the concept of common certainty. Something is common certainty for several people when it is true for them, but it is also true for them that it is true for the rest, it is true for all that it is true for all that it is true for all, and so on. With hats, the common starting certainty is that there is at least one white hat.
After the first round of responses, people come to have the common certainty that there are at least two white hats; if there were only one, the one who sees two black hats would have been able to answer the question. After the second round, the common certainty is that all hats are white, because if someone had seen a white hat and a black one, they would know that theirs is white.
As Robert Aumann demonstrated (and earned him a Nobel Prize in Economics), common certainty helps us resolve disagreements. In his famous agreement theorem, two people start with the same beliefs, then each acquires private information and, based on it, places a bet on an event. The theorem states that if the bets are common certainty, then they are equal.
The bets correspond to the probability that we assign to something happening – if we think it is very probable, we will bet more -, so the agreement theorem says that, if there was disagreement about the probability of an event, but that disagreement becomes in common certainty, everyone ends up agreeing.
It is clear that, in real life, many times the circumstances do not exist for this theorem to be fulfilled. It is not always based on the same information, nor are bets reviewed when they are common certainty and, above all, people are not always totally rational when deciding how much to bet. But the agreement theorem serves to provide internal coherence to any system where it can interact in a rational way. It is a key theorem in economics.
However, if we consider quantum systems the situation could be different: even starting from the same information and having common certainty, could we remain in disagreement? Quantum objects look different depending on how you look at them. For example, particles can be entangled in such a way that if you know something about one particle, the properties of the others are altered. In these kinds of situations, how could we reach a consensus? Is the agreement theorem still true?
Fortunately yes, as shown in a recent result. Although quantum theory is famous for creating conflicts with our intuition, it maintains the internal coherence given by the agreement theorem. Starting from the same information and having common certainty, it is still possible to reach an agreement, even if the entanglement is at the base of the communications used. This may be relevant in the not too distant future, in which the entangled particles will be a support for our communications and our finances.
Furthermore, in the aforementioned work, it is proposed to consider the agreement theorem as a physical principle. Thus, if in the future we develop a new theory that surpasses the quantum one in terms of its ability to explain the world we live in, but does not fulfill the agreement theorem, it is argued at work that we should discard it. In this way, we would make sure that any theory that we take for valid would fulfill this theorem, and, therefore, would have internal coherence.
This principle is added to many others that have already been proposed with the same objective: to reject theories that do not comply with them. Perhaps, one day, we will complete the list and see that quantum theory is the only one that fulfills all the principles. If we succeed, we will have found the best possible description of nature.
Patricia Contreras Tejada She is a doctor in quantum technologies and a science communicator.
Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which the researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Agate A. Rudder G Longoria (ICMAT).
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