A graph-theoretic explanation of Born's rule

Abstract: Arguably, the "right view" in science is the one from which one learns things that remain hidden in other ways to look at nature. There are many ways to look at quantum theory (QT). Each of them is organized around a particular fundamental concept, e.g., "quantum system", e.g., "measurement scenario". The problem is that none of these concepts is given a priori, but all of them are defined through the operations the experimenter may perform, and the experimenter can always enlarge at will both the quantum system and the scenario. Consequently, any general probabilistic theory (GPT) that provides probabilities to the measurement outcomes should consistently connect the probabilities of all these extensions. Therefore, a reasonable question is whether there is an alternative way to address the problem of what probability assignments a theory allows, without separating the study into quantum systems or measurement scenarios, but adopting a perspective that naturally contains them all.

Here, we will review an approach, the graph-theoretic approach to GPTs, that does precisely that. This nonstandard way to look at QT will turn out to be particularly illuminating about what is the property of nature that enforces QT.

For years, we thought that we would understand QT when we identify the physical principle that singles out QT from the set of GPTs. Behind this way of thinking is the assumption that there are logically consistent GPTs that can produce probabilities beyond those allowed by QT. The particular way to look at QT we will describe here shows that this is not the case, as it shows that there is a crucial message in the specific way nature is contextual: The sets of probability assignments to the outcomes of ideal measurements for every graph of exclusivity are indistinguishable from the ones of a universe with no laws. Therefore, from this perspective, QT is the largest and wildestlogically consistent theory. This is quite disturbing, as it suggests that the hypothetical principles singling out QT that we have been searching for do not exist, and that the ontological message of QT is that nature has no laws governing the outcomes of certain slicings of the world.