Zureck's Quantum Darwinism Paper

[lucas_]

None.

Thanks
Bill

What? so what interpretation is the Gleason theorem compatible with? It says in Wikipedia : "It proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space"... if none.. what kind of interpretation has to be made to make it compatible with it?

 

[bhobba]

What? so what interpretation is the Gleason theorem compatible with? It says in Wikipedia : "It proves that the Born rule for the probability of obtaining specific results for a given measurement follows naturally from the structure formed by the lattice of events in a real or complex Hilbert space"... if none.. what kind of interpretation has to be made to make it compatible with it?

Here is what it says without the advanced mathematical language of lattices etc.

By definition a Von Neumann observation is described by a resolution of the identity Ei such that outcome i is determined by Ei. To get the usual observable of an observation you associate a real number with each outcome to get O= Σyi Ei. Via the spectral theorem the two are equivalent.

What Gleason's theorem says is if the outcome can have a probability associated with it (its pretty hard to see how that would not be the case - but it is an assumption), and that probability only depends on the Ei (this is the assumption of non contextuality and is in fact the key assumption) then a positive operator of unit trace, P, by definition called the state of the system, exists such that the probability of outcome i is Trace(PEi).

Here is the detail:
http://kiko.fysik.su.se/en/thesis/helena-master.pdf

Born's rule easily follows E(O) = Σ yi * Prob outcome i = Σ yi *Trace(PEi) = Trace (P Σ yi* Ei) = Trace (PO).

This is part of the standard quantum formalism so is applicable to all interpretations.

Just a comment. You are jumping to all sorts of erroneous conclusions without thinking things through. There was nothing in the Wikipedia article that said it wasn't applicable to all interpretations. You came across some terms like lattice and jumped to a conclusion without understanding what lattice means here. Can you please try to be more careful?

Thanks
Bill

 

[atyy]

What? so what interpretation is the Gleason theorem compatible with?

If you read your question and bhobba's answer carefully, you will find that he said that Gleason's theorem is compatible with all interpretations.

 

[naima]

So quantum Darwinism tried to explain why pointer states are classical.. it still doesn't explain why there is definite outcome (or why certain eigenvalues of the preferred basis is chosen)

Yes it does.

In decoherence we usually read that a pure density matrix evolves toward a mixed one diagonal in a basis and that one of the vectors of this basis will be the output of the measurement.

Here Zurek reverses the way to see things. When a system decoheres while interacting with the environment it leaves many imprints in it. Eg take a grain of dust in a superposition of what you want. When it is illuminated it returns many photons in different directions. Many observers can receive them. they will agree that they come from a same source. An observable emerges from the decoherence Zurek calls it $\pi$ and writes:

Observers probing fraction of the universe can act AS IF the system has a state of its own (one of the eigenstates of $\pi$)

 

[TrickyDicky]

... he said that Gleason's theorem is compatible with all interpretations.

I 'm not seeing this. Gleason's theorem assumes non-contextuality, which is a strong assumption, hard to reconcile with the practice of QM and a reality that appears to be contextual, But my point is that interpretations like say Bohmian, or even some Copenhagen flavors include contextuality, how is Gleason's non-contextuality assumption compatible with that?

 

[atyy]

I 'm not seeing this. Gleason's theorem assumes non-contextuality, which is a strong assumption, hard to reconcile with the practice of QM and a reality that appears to be contextual, But my point is that interpretations like say Bohmian, or even some Copenhagen flavors include contextuality, how is Gleason's non-contextuality assumption compatible with that?

In BM, the Copenhagn interpretation is emergent, so BM is consistent with Copenhagen.

 

[TrickyDicky]

In BM, the Copenhagn interpretation is emergent, so BM is consistent with Copenhagen.

Be that as it may, how does it explain their contextuality being compatible with Gleason's theorem non-contextual assumption?

 

[atyy]

Be that as it may, how does it explain their contextuality being compatible with Gleason's theorem non-contextual assumption?

I don't know how to answer this in a short way, but if BM is comptaible with standard QM, and standard QM is compatible with Gleason's then BM is compatible with Gleason's. It has to do with the definition of "measurement" in QM, and how "measurement" is implemented in BM via decoherence and he way a preferred basis is continuously picked out by the choice of hidden variable and Bohmian dynamics.

 

[TrickyDicky]

I don't know how to answer this in a short way, but if BM is comptaible with standard QM, and standard QM is compatible with Gleason's then BM is compatible with Gleason's. It has to do with the definition of "measurement" in QM, and how "measurement" is implemented in BM via decoherence and he way a preferred basis is continuously picked out by the choice of hidden variable and Bohmian dynamics.

It's the part about measurement in standard QM (that I am identifying here with Copenhagen) compatibilty with Gleason's theorem that I'm not getting, I can see how it is compatible with the formalism of standard QM without measurement-state reduction.

 

[kith]

I also don't see a problem here. dBB is contextual, so Gleason's theorem doesn't apply. You don't need to reconcile the contextuality of dBB with the non-contextuality assumption of Gleason's theorem. I would say that in the context of dBB, you simply can't use it to derive the Born rule.

And if I get dBB right, there is indeed a very different justification of the Burn rule there: Similar to the 2nd law in statistical mechanics, the Born rule is very likely to hold but it isn't a strict condition. See https://en.wikipedia.org/wiki/Quantum_non-equilibrium

 

[atyy]

It's the part about measurement in standard QM (that I am identifying here with Copenhagen) compatibilty with Gleason's theorem that I'm not getting, I can see how it is compatible with the formalism of standard QM without measurement-state reduction.

State reduction can be derived in BM.

 

[TrickyDicky]

Ok, thanks. I simply cofused myself about what "compatible" implied above.

 

[atyy]

Ok, thanks. I simply cofused myself about what "compatible" implied above.

Thinking about it, I like kith's reply in #40 better. I don't know if contextuality can be derived in BM.