How quantum mechanics requires non-additive measures

This video shows how a quantum analogue of the count of states must be non-additive: the total number of states is not necessarily the sum of disjoint subsets. The non-additivity appears because of contextuality: not all states are counted at all-else-being-equal. Quantization can be understood as imposing a count of states that is finite on finite continuous sets and is unitary on each single state.

The ideas presented are based on the following paper:

[mikoajmetelski]

Didn't Groenewold figure out the phase space representation of quantum mechanics? I would have thought that was the framework in which positivity of the (quasi)probability density is sacrificed instead of additivity... but not sure how that relates to the whole picture

[claudioalbanese]

This line of arguments is interesting but it can be simplified without recourse to sigma algebras and Lebesgue measures which give rise to experimentally undecidable statements and should be out of the scope of this line or research. 

Both classical mechanics and quantum mechanics can be described as the limit of numerical discretisation schemes based on finite floating point precision and in both cases we can pass to the continuous time limit to recover the usual limits that make use of real numbers. It will suffice to work with an element in the approximation sequence, notwithstanding the numerical errors. In this context, all statements are decidable in finite time and we can place an upper bound on the time required to make a decision.

Classical mechanics and classical statistics can be defined as the realm where the Bayes rule for conditional expectations is satisfied. If we have a system and we can probe it with two instruments and find the value of two distinct observables, then the result of performing two observations one after the other is consistent with the Bayes rule of conditional expectations. Mathematically, this implies that a pure state in a classical context has a certain value of all the observables. It makes sense to ascribe a positive number between 0 and 1 to any pure state. Furthermore pure states form a basis for the linear space of all probability distributions (which in general describe mixed states). 

In the domain of quantum mechanics, Bayes' rule of conditional probabilities fails. If we prepare a spin in a certain direction and then measure it's spin first in the x and then in the z direction and then repeat the experiment making measures first in the z and then in the x direction we see that the classical law of conditional expectations does not apply.

A first consequence of this experimental evidence is that in quantum mechanics we must allow for pure states where the set of observables does not have univocal values. But is this just a question of resolution? Does there exist an underlying classical system while our ability to prepare states is limited and restricted to some family of mixed states? Or is there something deeper going on? QM says that there is something deeper going on and we have to take the square root of the classical concept of additivity.

Namely, QM replaces classical additivity with the concept of coherent superposition involving measurements on the superposition of two pure states. A coherent superposition is an entangled state and we know experimentally that it can be prepared. It so turns out that by taking the linear superposition of wave functions and then calculating probabilities as norms squared, we end up with results in line with experiments.

This is not the only instance of QM reality wanting us to take square roots. Also the Dirac equation is a square root of KG and explains how half integer spins arise, a purely quantum concept. In a sense, QM is the square root of classical mechanics and it becomes relevant when experiments indicate that observables do not commute.