Where can we bend the rules of QM?

This is a plea for the experts to refine my understand:

As I understand it, the current state-of-the-art on reconstruction of
QM from postulates, in the vein of von Neumann, has gotten so far:

1. The fundamental measurement of a system, or possible information
about a sytem, is a binary valued question.
2. Each system has a finite number of such questions which maximally
describe it.
3. There are still more questions, the answers to which are not
determined from a maximal description.

I believe that it is a theorem that this gives an orthomodular lattice
containing subsets which are boolean algebras. Now the next step is
hazy: that it implies a representation as a Hilbert space with
projectors. Words seem to be minced over this point. Is this really
true? Is it only true if we understand "representation" to be some
homomorphism to a vector space? If so, why a vector space?

Now, we assume that it is a vector space (Piron suggests that a
Hilbert space over R, C or H would do). Following Chris Fuchs, we
introduce another postulate,

4. The answer to a question is independent of the other questions
asked -- non-contextuality.

Then we recover the tensor product structure for combining systems
together, POVM for measurements, density matrices and the Born rule
for interpretation as probabilities. As a corollary, we exclude R and
H for the field over which our Hilbert space is defined, as only C
gives the tensor product rule uniquely; to be rigorous, we've only
excluded H as it over-constrains the choice -- R would need some other
postulate.

More speculatively, we can recover unitary time evolution, if we're
willing to accept:

5. It is not possible to physically perform NP-complete problems in P
time.

Question for the experts of SPR:
Is this understanding correct? And *why* on earth a vector space?

Regards and thanks,
Gen Zhang

.

Relevant Pages