Philip Ball has just published an excellent article in the Quanta magazine about two recent attempts at understanding the Born rule: one by Masanes, Galley, and Müller, where they derive the Born rule from operational assumptions1, and another by Cabello, that derives the set of quantum correlations from assumptions about ideal measurements.
I’m happy with how the article turned out (no bullshit, conveys complex concepts in understandable language, quotes me ;), but there is a point about it that I’d like to nitpick: Ball writes that it was not “immediately obvious” whether the probabilities should be given by $\psi$ or $\psi^2$. Well, it might not have been immediately obvious to Born, but this is just because he was not familiar with Schrödinger’s theory2. Schrödinger, on the other hand, was very familiar with his own theory, and in the very paper where he introduced the Schrödinger equation he discussed at length the meaning of the quantity $|\psi|^2$. He got it wrong, but my point here is that he knew that $|\psi|^2$ was the right quantity to look at. It was obvious to him because the Schrödinger evolution is unitary, and absolute values squared behave well under unitary evolution.
Born’s contribution was, therefore, not mathematical, but conceptual. What he introduced was not the $|\psi|^2$ formula, but the idea that this is a probability. And the difficulty we have with the Born rule until today is conceptual, not mathematical. Nobody doubts that the probability must be given by $|\psi|^2$, but people are still puzzled by these high-level, ill-defined concepts of probability and measurement in an otherwise reductionist theory. And I think one cannot hope to understand the Born rule without understanding what probability is.
Which is why I don’t think the papers of Masanes et al. and Cabello can explain the Born rule. They refuse to tackle the conceptual difficulties, and focus on the mathematical ones. What they can explain is why quantum theory immediately goes down in flames if we replace the Born rule with anything else. I don’t want to minimize this result: it is nontrivial, and solves something that was bothering me for a long time. I’ve always wanted to find a minimally reasonable alternative to the Born rule for my research, and now I know that there isn’t one.
This is what I like, by the way, in the works of Saunders, Deutsch, Wallace, Vaidman, Carroll, and Sebens. They tackle the conceptual difficulties with probability and measurement head on3. I’m not satisfied with their answers, for several reasons, but at least they are asking the right questions.