atyy said:
Actually, why do you say PVMs are more fundamental? One of the questions I'm interested in for this thread is whether there is any "physical" argument that would make PVMs and projective state reduction more fundamental than POVMs and completely positive trace non-increasing maps. From a Copenhagen viewpoint, and I think also a Bohmian viewpoint, POVMs and CP maps seem to be more fundamental. From a Copenhagen viewpoint, I think one argues that the probability distributions we can observe should respect the vector space structure, and the state reduction has to be completely positive so that if we extend the system, the state reduction of the extended system also takes density matrices to density matrices.
There is a good traditional physical argument for the von Neumann projective measurement in that it is repeatable, which is clearly a good condition for the pointer. But I don't know if this can be extended to continuous variables. It can be argued that we only need large but finite dimensional Hilbert spaces (especially since the standard model is probably not UV complete), in which case the von Neumann measurement is certainly fundamental. But here I would like to consider what happens if we also allow infinite dimensional Hilbert spaces and continuous variables to be fundamental in quantum mechanics.
Yes, the uniqueness of the orthogonal decomposition (when considering system+apparatus+environment) is why I thought maybe decoherence provides a physical viewpoint in which one could argue that PVMs and projective state reduction are more fundamental. Is that right, or can decoherence lead to a POVM (that is not physically derived from a PVM)? Another thought is that even if it is right that decoherence favours PVMs as fundamental, it is not very physical, since there is no exact decoherence?
Let me put it this way.
If you think that quantum physics is fundamentally about (i) quantum formalism and (ii) classical macroscopic apparata (which is a rather Copenhagen view), then you are right, POVM is more fundamental.
But if you think that ALL nature is fundamentally quantum, then POVM cannot be fundamental. Let me explain. If all nature is fundamentally quantum, then you want to EXPLAIN why macroscopic world LOOKS classical, despite the fact that it is really not classical. (In the Copenhagen approach you do not try to explain it, you just take it for granted.) To explain it, you need a quantum theory of measurement which takes into account not only the quantum state of the microscopic measured system (which is what a Copenhagen approach does), but also the quantum state of the macroscopic measuring apparatus (which is typicaly ignored in a Copenhagen approach). When you do that, then the role of decoherence becomes essential. You are right that there is no exact decoherence and hence no exact PVM in such an approach. Yet, the PVM description of such systems is an almost perfect approximation valid FAPP. Moreover, the validity of such an approximation is essential for the decoherence approach to work. If you had POVM's which could not be well approximated by PVM's, then the decoherence approach would not work and you could not explain how the quantum measurement works and why macroscopic world looks classical despite the fact that it is really quantum.
Concerning continuous variables, in practice you can never measure a continuous spectrum. But you can measure (and do measure) a pseudo-continuous spectrum, that is, a discrete spectrum with a very small difference between values that can be experimentally distingushed. Here it is important to stress that in a non-Copenhagen approach it does not mean that discreteness is fundamental, because fundamentally (in a non-Copenhagen approach) quantum mechanics is NOT only about measurements.
In fact, in a non-Copenhagen approach neither PVM nor POVM are truly fundamental. Both are tools for describing measurements, and measurement is not fundamental in a non-Copenhagen approach. But PVM is more IMPORTANT than POVM, because it plays an important (even if only approximate) role in explaining the classical appearance of the quantum world at the macroscopic level.
I am not sure that this removes all your doubts, but I hope that it still helps. If you have further questions, I can try to answer them too.