Why doesn't <PERSON>'s theorem imply the <PERSON> rule?
I know that the question "does <PERSON>'s rule follow from <PERSON>'s theorem" has already answers on the website: see here, and here. I am not satisfied with the answers given (one cannot rule out that I have not understood them) and think my question is not a duplicate (if it is, please show me).
I think that a (pretty obvious and rather trivial) derivation of <PERSON>'s rule can be made from <PERSON>'s theorem and a few assumptions; so, I think it would be easy for you to point out which assumption you would reject, based on physical or philosophical reasons.
Let us consider a fixed system.
1) Assume that a (finite dimensional of dimension greater or equal than $3$) Hilbert space $\mathcal{H}$ is given.
2) Assume that for every measurement device that has output set a finite set $S$, a family $(H_s)_{s \in S}$ of pairwise-orthogonal subspaces of $\mathcal{H}$ is given, called the eigenspaces of the measurement device.
3) Assume that every finitely-indexed family of pairwise-orthogonal subspaces of $\mathcal{H}$ is associated to a measurement device.
4) Assume that to each measurement device with output set $S$, is assigned a family $(p_s)_{s \in S}$ (that are thought as probabilities) of real, nonnegative real numbers that sum to $1$.
5) Assume that whenever a subspace $H$ of $\mathcal{H}$ appears as an eigensubspace of two measurement devices, the probabilities in 4. are equal.
Consider, for each subspace $H$ of $\mathcal{H}$, define $\phi(H)$ to be the probability given by 4. to the corresponding output of any measurement device that has $H$ as eigenspace (there is at least one, from assumption 3.). I claim that this map $\phi$ is well-defined, because of 5., and from 3., it satisfies the hypotheses of <PERSON>'s theorem.
Indeed:
* Let $H$ a subspace. If $H$ is an eigenspace of two measurements devices, then the assigned probabilities are the same. So $\phi$ is well-defined.
* For every subspace $H$, $\phi(S) \in [0,1]$ by definition.
* Let us prove that $\phi$ is additive. Let $(H_1,\cdots,H_n)$ be a family of pairwise-orthogonal subspaces. We have to show that $\sum^n_{i=1} \phi(H_i) = \phi(\oplus^n_{i=1} H_i)$. 1020
Let $H := \oplus^n_{i=1} H_i$. For that, consider two measurement devices: the first has $\left(H_1,\cdots,H_n,H^\perp\right)$ as eigensubspaces, and the second has $\left(H, H^\perp\right)$. According to 4. and 5., $1 = \phi(H^\perp) + \sum^n_{i=1} \phi(H_i)$ and $1 = \phi(H) + \phi(H^\perp)$. Therefore, $\sum^n_{i=1} \phi(H_i) = \phi(H)$.
Therefore, there is a unique density matrix $A$ such that for each subspace $H$, $\phi(H) = Tr(AP_H)$; so, we have <PERSON>'s rule.
I think that 1. and 2. are not the core of the problem; 3. is a technical assumption, and some strengthening of <PERSON>'s theorem could allow a weaker assumption (but this is a math problem); 4. can be thought as an experimental assumption (these probabilities can be given as limits of experimental frequencies) or an ontological assumption; to me, the only assumption that stands out is 5 and I feel it is the only one that one should reject, but I am not sure everybody thinks this way.
My question is: can you indicate me which of these assumptions is controversial enough to make people think that <PERSON>'s rule does not follow from <PERSON>'s theorem? Why? Or do you think this argument is missing the point?
EDIT: I forgot an important point. 680