Relational Interpretation of QM

[alexepascual]

I have started to read about Rovelli's Relational interpretation in Wikipedia.
http://en.wikipedia.org/wiki/Relational_quantum_mechanics#All_systems_are_quantum_systems
In the section labeled "All systems are Quantum Systems" in the article it says:

Taking the model system discussed above, if O' has full information on the system, he will know the Hamiltonians of both S and O, including the interaction Hamiltonian. Thus, the system will evolve entirely unitarily (without any form of collapse) relative to O', if O measures S. The only reason that O will perceive a "collapse" is because he has incomplete information on the system (specifically, he does not know his own Hamiltonian, and the interaction Hamiltonian for the measurement).

In the above paragraph it is assumed that both S , O and O' are two-state systems. O measures S and some time later O' will measure (S+O). I will label them "+" and "-". What I don't understand is the claim that when O measures S, their state after measurement can be seen as a unitary evolution from the state before measurement. I understand that to O' the composite system will be in a superposition |O+,S+> and |O-,S-> before it decides to measure it. But this superposition means a reduction of the product Hilbert space because now the states |O+,S-> and |O-,S+> can't be observed by O'.
It seems to me that even though O' does can't assume a total collapse, the establishment of entanglement between O and S implies a non-unitary (projective) transformation.
Any ideas about this?

 

[ConradDJ]

From the point of view of O', you're right that there is a difference between the states of O and S prior to their interaction, and their entangled state after the interaction. When O' gets the information that O has measured S, for O' there's a discontinuous change in the state of the combined O-S system.

But at this point there is still no difference in the probability that S (or O) will be found in state - or state +. So, if I understand this correctly, for O' there is no reduction in the state of these systems considered separately.

 

[alexepascual]

From the point of view of O', you're right that there is a difference between the states of O and S prior to their interaction, and their entangled state after the interaction. When O' gets the information that O has measured S, for O' there's a discontinuous change in the state of the combined O-S system.
But at this point there is still no difference in the probability that S (or O) will be found in state - or state +. So, if I understand this correctly, for O' there is no reduction in the state of these systems considered separately.

Once O and S have interacted, O' should not think of them as being on separate Hilbert spaces right? But I think that in the product Hilbert space, the entanglement process can't be described by a unitary transformation. If I understand the article correctly, it claims it does change unitarily. This is what I am objecting to.
You are saying that there is a discontinuous change. I understand that the projection process during measurement is usually thought of as discontinuous. I wouldn't streess too much on the discontinuous nature as it may turn out that looking at it with a better time resolution it is more or less continuous. But my objection is that some unitary transformation should map all the vectors in Hilbert space to the same Hilbert space (all of it I think).
In this case, the transformation maps all vectors in Hilbert space (the compound Hilbert space) to a sub-set of it. This tells me that the transformation is not unitary because it does not preserve volume. Or I am mistaken or the article in Wikipedia is wrong. I would not think that Rovelli or people in their group could be wrong because they are obviously tremendously knowledgeable about the subject. But there is a possibility that the person who wrote the Wikipedia article made a mistake.
Here is the link to the article:
http://en.wikipedia.org/wiki/Relational_quantum_mechanics#All_systems_are_quantum_systems

 

[ConradDJ]

Hi -- I was hoping for a response from someone who understands this better than I do, but I think you are probably right and the Wikipedia article is not accurate. It seems to be a work in progress, to be generous. Anyway I recommend you look at Rovelli's initial paper (pg 17 top right-hand column is Rovelli's version of this).

http://arxiv.org/abs/quant-ph/0604064v3"

 

[ConradDJ]

Sorry, that was the link to his EPR paper. I meant pg 17 of this one:

http://arxiv.org/abs/quant-ph/9609002v2"

Conrad

 

[Fra]

I'm not sure if there was a problem here related to the relationa interpretation or what was the problem. I might pick on several other things, but which are more subtle.

But maybe you questioned wether the so called pre-measurement is unitary or not? Note that O' does not actually perform a measurement in the second description. It's the time evolution when O' describes S-O as per the schrödinger equation, and this is a regular schrödinger type unitary time evolution. The idea is that pre-measurement is effectively a kind of (unitary) diagonalization, carried out by the physical process implicit in the hamiltonian operator (not written out, because it's just an example of principle).

Does that help?

/Fredrik