Is the QM axiom for measurements misleading?

[vanesch]

Again, I'll point out MWI as a counterexample, since it does not make that assumption. Probabilistic predictions are made through other means. (and, I think, how to get probabilities is one of the main topics of research in MWI. Maybe vanesch can clarify if he appears)

It is in fact the "holy grail" of MWI, as indeed, there doesn't seem to be an obvious way now to generate probabilities. There are different MWI flavors and they often diverge on the details on how to see probability.

However, there is one red line through all these views: that is: instead of QM generating probabilities for events, it generates probabilities for observers.

That is, for each thing we call a "measurement", it is a measurement wrt to an observer, that is, a psycho-physical link to a subjective experience (that is what is, ultimately, an observer). You have to make a distinction between the "physical degrees of freedom" of the material carrier of an observer, and the observer itself (which is a subjective experience that goes with it). For a human, for instance, the physical degrees of freedom correspond to the quantum mechanical description of his body. Now, if these physical degrees of freedom occur in different terms in the wavefunction, which are entangled with other systems and have a certain stability in time and all that (it is on these points that different views on MWI differ), then that means that to the same set of physical degrees of freedom, can be associated different observers ("copies of them"), and the wavefunction generates in one way or another (again, different views here correspond to different flavors of MWI) a statistical ensemble of "observers".

In other words, by simply applying the Schroedinger equation, you arrive at a state which can formally be represented as:

|my-body-sees-dead-cat> |dead-cat>|stuff> + |my-body-sees-live-cat>|live-cat>|otherstuff>

The physical degrees of freedom of my body occur in two terms, entangled with other stuff. This generates a statistical ensemble for subjective experiences ("observers") to be attached to my body. MY personal subjective experience being one of them, I draw my subjective experience from that ensemble, which gives me the probabilistic impression.

Now, it is mostly disturbing to have to talk about "subjective experience" in a physical theory. This is what turns off most people from the start in MWI. But we have also to realize that the only way of knowing that there is in fact a kind of measurement problem in QM, is through subjective experience ! If it weren't because we "know" that we cannot be "at the same time" walking in the park and shopping in the grocery store, by our experience, then this would not be contradictory! All things you could calculate to both of these states in superposition would still be correct (they would happen then "simultaneously"). So it seems that the only objection against superposition comes from our subjective experience - in that case one shouldn't be surprised that it is also part of the proposed solution.

EDIT: let me clarify a bit more the last bit.
We have a serious problem in considering "ourselves in superposition". Nevertheless, if we say that we are to have a quantum description (a vector in hilbert space), then it should be obvious that the superpositions of "ourselves here" and "ourselves there" is entirely part of the description. So the problem is already introduced from the very start: if our bodies are to have a genuine quantum description, there's no way to exclude these superpositions.
You might hope somehow that some dynamical instability will remove these states, and quickly bring them to the "me here" state, or the "me there" state, but unfortunately, with a unitary time evolution, that's impossible to achieve (THIS is the fundamental problem):
if some initial state leads to "me here" and another initial state will lead to "me there", then the superposition of these two initial states will lead to a superposition of "me here" or "me there".

So logically, there are only 3 ways out:
- quantum descriptions don't apply to things like human beings, or we don't have to take them literally. Ok, but then QM is not a universal theory, and the question is then: what is a universal theory ? This includes essentially all "shut up and calculate" views, who only see the quantum formalism as a technique that allows to calculate outcomes of observations, as well as alternatives to QM (local or nonlocal realist theories...).
- the unitary evolution is not strictly correct, and the corrected dynamics WILL include these kinds of instabilities. This is very well possible. GRW is one such approach ; Penrose thinks that gravity in one way or another plays this role.
- these superpositions really do occur. In that case we have to say why we don't experience life that way. That's essentially MWI.

EDIT II:

for a (very good) overview of the ideas behind MWI, look here:

http://plato.stanford.edu/entries/qm-manyworlds/

 

[vanesch]

If you think twice, you will realize that the widespread use of the density operator (rho) is precisely justified by this useful "interpretation". Generally, experimental prediction by a theoretical model are expressed by an average of "detection" operator with the famous formula: <D> = tr[D rho] . This could be for example the intensity of an atomic emission. You will also agree, I think, that quantum mechanics can be fully constructed on the density operator. And this would smooth out much of the debate.

The problem with the use of the density operator this way, is that you *already* made use of the projection postulate in order to write <D> = tr[D rho], when you apply it to a reduced density matrix of a subsystem.

I think it should be possible to prove the (practical) equivalence between two descriptions:

the entangled state description (macro-micro) (the full density matrix)
the statistical projection description (the diagonal part of the density matrix)​

Sure, that's the entire content of decoherence theory. But, it doesn't give a solution to the measurement problem as such. It only indicates that *if you have a solution to the measurement problem*, that in that case, you can place the Heisenberg cut anywhere, once there is entanglement with the environment, because this will not numerically change the outcomes.

If you are interested in these issues, I suggest you look at "Decoherence and the appearance of a classical world" by Zeh, Joos et al. (look on amazon for details).

But I didn't see why i would need a MW reality to understand QM.
I also understand by the discussion that the fathers of QM did their best with the projection postulate, but that it is indeed not absolutely necessary.

The density matrix formalism uses the projection postulate in a disguised way. If you have a pure state: |psi>, and you calculate rho = |psi><psi|, and write out the matrix in your preferred basis, then you have the corresponding "Born rule probabilities" on the diagonal. However, the matrix is not diagonal in all generality. "After the measurement", it has become a statistical mixture, and now we should put all the non-diagonal elements to zero. It is this operation which is mathematically impossible by a unitary time evolution. So no time evolution operator you can think off can do this.

However, what you can do is to "trace out the environment". That is, calculate for an entangled state, the reduced density matrix that corresponds only to a subsystem. What decoherence theory shows you, is that, with enough "irreversible entanglement in practice", that this reduced density matrix will take on a certain diagonal form in a "basis which is robust wrt to the environment" (which comes down to the so-called pointer basis).
If you now interpret this reduced density matrix as a genuine density matrix, then it looks like the density matrix of a mixture.
However, it isn't really a mixture! The genuine density matrix is the larger matrix, including the environment, and there exists in principle a unitary operator which can "undo" the entanglement - which wouldn't be the case if we had a genuine mixture. We can establish correlations with the overall density matrix, which are not described by the local reduced density matrix.
The local density matrix is in fact obtained by assuming that at some point, we can apply the Born rule on the larger system, and then sum over the probabilities of the "irrelevant" degrees of freedom of the environment. That's exactly what is done by taking the partial trace. If you don't assume that, then there is no meaning to "taking the partial trace".

 

[Anonym]

Lalbatros:”Remember the title of this thread.
My question was precisely about the collapse of the wave function and specifically about how the ad-hoc postulate has (mis-) led people to unrealistic discussions (by unrealistic I mean useless interpretations ...).”

The “collapse of the wave function” is not a postulate and it is not interpretation dependent. The collapse is universally valid firmly established experimental result connected with the transition from Quantum world to Classical world (E. Schrödinger cat). The collapse is a key problem of the measurement theory and therefore also is called “The Measurement Problem”. The theoretical description of the natural phenomena is groundless without consistent and adequate measurement theory. There are and always will be people that claim that a quantum world is a classical world or a classical world is a quantum world, that the measurement problem do not exist, AB phenomenon do not exist,quarks do not exist, etc. These are the people that have no problems in their worlds. God bless them. But it is not interesting world.

Lalbatros:
”That the SE governs quantum systems is not an interpretation.

The probability amplitude is not either, it is the root definition in QM.

And I agree that any interaction, was it with a beam splitter or with a photomultiplier, should be included in the relevant simulation of a complete experiment (or phenomenon).”

May be you are right. Then the interaction should be treated as the communication message. It should be encoded by transmitter and decoded by receiver. The transmitter code is the complex Hilbert space mathematical structure. The receiver code should be matched. And the classical physics are dispersion free. Therefore, the real Hilbert space will do a job.