What is the place of probability in quantum mechanics?

Question

I have some phenomenological problems with understanding probabilities in quantum mechanics, and I suspect that the reason for the confusion is that scientists themselves have not yet fully decided.

There are several options:

1. The first option, which goes back to the classics. In quantum theory itself and its dynamics, there are no probabilities. A quantum system evolves unitarily, if it is a system of two interacting particles, then it is a single system, the particles are in a state of interaction constantly and continuously, that is, it is impossible to factorize the state. The probability appears only when an external macroscopic observer, who does not have a complete description of the system, makes a measurement. That is, the very concept of probability concerns the relationship between a quantum system and a macroscopic observer.

2. Everything is probabilistic. Even without any measurement, two quantum particles interact probabilistically, that is, after passing at a certain distance from each other, they will either interact or not (depending on the probability).

Which point of view is the most correct?

Answer 1

QM is intrinsically a probabilistic theory: It says nothing about single systems as it deals with ensembles of systems. Therefore the notion of probability is pervasive. The notion of state itself has a probabilistic nature. A state is nothing but the assignment of the probability of every elementary YES-NO proposition testable on the systems of the ensemble. Such an assignment is equivalent to a density matrix or a state vector as proved by Gleason with his celebrated theorem. Pervasivity is also evident, for instance, in the use of unitary operators to describe symmetries but also time evolution is a consequence of the requirement that probabilities are preserved under the action of the symmetry or the time evolution.

Classical physics can be stated in a completely probabilistic fashion as well. A state is a Liouville probability density in the space of phases and sharp states are described by Dirac deltas, which are probability measures as well. From this perspective QM and CM are quite close to each other. The difference stays in the space of events where the notion of probability is given which is different in classical and quantum theory to take account, in the second case, of the existence of incompatible (in quantum sense) observables. This answer of mine tackles these issues from a general perspective

Answer 2

It all depends on your interpretation of QM. Your case 1. seems to come close to relational quantum mechanics. In RQM we cannot ask “what state is Schrödinger’s cat in ?” (well, we can ask, but the question has no answer). All we can ask is “how can we best describe our knowledge about Schrodinger's cat ?”. Before the box is opened our best description is a superposition of “alive” and “dead” states, and we can create a wave-function that represents this superposition. After the box is opened, we know that the cat is alive (or maybe dead) but this is a change in our knowledge about the cat, not the mysterious physical collapse of some wave-function.

Answer 3

You are not alone in worrying that the Born rule seems like an ad hoc part of quantum mechanics. The part that physicists should agree on is that option 2 is not correct because it's very similar to the premise of a hidden variable theory. At least when there is no observation occurring, a system's wavefunction should evolve unitarily.

Within option 1, however, there are conflicting interpretations because performing a measurement on any system inevitably means entangling it with our measurement apparatus. So the question is what exactly happens during this decoherence. I hope I'm not being unfair to them when I say that Copenhagen adherents believe there is a process somewhere along the line which is not unitary evolution. This is a collapse where one outcome in the support of the wavefunction has been "chosen" and the rest "forgotten".

Personally, I find the Many Worlds interpretation more appealing which holds that the rest of the wavefunction still exists. This means that there is no collapse when we open the box containing Schroedinger's cat. Instead, opening it just ensures that the state of our eyes and the state of the cat can no longer be factored and we have to talk about a joint wavefunction where they are highly correlated. It's still unclear why we only perceive a part of this wavefunction instead of the whole thing. But in Many Worlds, that becomes a question about consciousness instead of physics.