Zureck's Quantum Darwinism Paper

[bhobba]

Hi Guys

Quantum Darwinism came up in another thread so I thought I would go through Zurecks paper on it.
http://arxiv.org/pdf/0903.5082v1.pdf

Went through it years ago but know quite a bit more about the foundations of QM now.

What struck me this time was his postulates:

'Quantum physics is based on several “textbook” postulates The first two; (i) States are represented by vectors in Hilbert space, and; (ii) Evolutions are unitary – give complete account of mathematics of quantum theory, but make no connection with physics. For that one needs to relate calculations made possible by the superposition principle of (i) and unitarity of (ii) to experiments. Postulate (iii) Immediate repetition of a measurement yields the same outcome starts this task. This is the only uncontroversial measurement postulate (even if it is difficult to approximate in the laboratory): Such repeatability or predictability is behind the very idea of “a state”. In contrast to (i)-(iii), collapse postulate (iv) Outcomes correspond to eigenstates of the measured observable, and only one of them is detected in any given run of the experiment, is inconsistent with (i) and (ii). Conflict arises for two reasons: Restriction to a preferred set of outcome states seems at odds with with the egalitarian principle of superposition, embodied in (i). This restriction prevents one from finding out unknown quantum states, so it is responsible for their fragility.'

My concern has to do with Gleason's theorem. Starting from Von Neumann measurements (the probability of outcome i is described by a resolution of the identity Ei, such that outcome i depends only on Ei) one applies Gleason and you get; first, states must exist (and they obey by fiat the principle of superposition since pure states form a vector space), and secondly, the Born rule he asserts is in conflict with the first. That however is logically not possible. I am not worried about the unitary thing since that falls out of Wigners Theorem.

Really scratching my head here.

What do others think?

Thanks
Bill

 

[atyy]

He's just stating things very loosely. He can state it more properly, but he knows and his audience knows that the problems don't go away, so he sticks to the naive form of the postulates in the Copenhagen interpretation for simplicity. Because of the Born rule, in order to describe successive measurements in the Schroedinger picture, the state evolution must be random after after the Born rule has been applied when a measurement is made. Since there are two rules of time evolution - one deterministic and one random, in order to avoid a contradiction, they must be applied at different times. An external observer is needed to decide when a definite outcome is obtained, which is the measurement problem.

 

[bhobba]

Hi Atty

Mate I think you nailed it.

I had to take a trip since I posted it and thought about it during the drive and that's the conclusion I pretty much reached.

Thanks
Bill

 

[lucas_]

Bill. After reading Zurek quantum Darwinism papers for 5 years.. I still can't understand how it is connected to Copenhagen so let me ask this so it's clear up once and for all. I know it is about (Maximillian page 86) "Thus the role of environment is now broadened, namely, from the selection of preferred states for the system of interest and the delocalization of local phase coherence between these states to the transmission of information about the state of the system. The key question, first spelled out by the Los Alamos group, is then the following. How, and which kind of information is both redundantly and robustly stored in a large number of distinct fragments of the environment in such a way that multiple observers can retrieve this information without disturbing the state of the system, thereby achieving effect classicality of the state?"

Why does Zurek worry about how multiple observers can retrieve this information without disturbing the state of the system. For example. You measure the electron using the position operator while atyy measure it using the momentum operator. You two will get different answers. This is commonly done in laboratory so what's the problem. So why propose quantum Darwinism where you and atyy has to both read the same state of the electron? This is the reason if I thought this had to do with macroscopic object like window as we discussed. And you two said it didn't.. so for microscopic object, what's the deal if you two read different results since you use different instrument (and hence shift the operators)?

 

[bhobba]

Why does Zurek worry about how multiple observers can retrieve this information without disturbing the state of the system

Think to Schroedingers Cat. You can tell if the particle detector clicked by looking at if the vial was broken or if the cat was alive or dead. They are different observations, part of the particle detectors environment - different fragments as he says. But they should all agree - which is a central concern he has.

Thanks
Bill

 

[lucas_]

Think to Schroedingers Cat. You can tell if the particle detector clicked by looking at if the vial was broken or if the cat was alive or dead. They are different observations, part of the particle detectors environment - different fragments as he says. But they should all agree - which is a central concern he has.

Thanks
Bill

You told me the window won't change by each person looking at it because it's locked to the eigenstates of positions. So it really can change (without quantum Darwinism)? But how does it and the cat eigenstates of positions change (hence changing the window shape)? Why didn't you say so, I was perplexed by it for a week.

 

[atyy]

After reading Zurek quantum Darwinism papers for 5 years.. I still can't understand how it is connected to Copenhagen so let me ask this so it's clear up once and for all.

Yes, Quantum Darwinism does make sense in Copenhagen. In Copenhagen, one can place the classical/quantum cut at different locations along the von Neumann chain of different observers and get consistent results. Decoherence and Quantum Darwinism show this consistency.

 

[lucas_]

Yes, Quantum Darwinism does make sense in Copenhagen. In Copenhagen, one can place the classical/quantum cut at different locations along the von Neumann chain of different observers and get consistent results. Decoherence and Quantum Darwinism show this consistency.

without quantum darwinism, you would see a cat sleeping and bhobba would see the same cat that is jumping up and down? can you give other examples for macroscopic object if quantum darwinism didnt exist.. like would you literally see window that is square while bhobba would see a triangle window? or is it all valid only for quantum object in coherence? but bhobba mentioned a cat...

 

[bhobba]

You told me the window won't change by each person looking at it because it's locked to the eigenstates of positions.

I didn't say that.

I said in my interpretation its not an issue because I put collapse immediately after decoherence and everything is classical after that. And I specifically pointed out Zureck has a different view and you have to read his paper to understand it rather than the limited account in Schlosshauer.

Again, as I said, there is a tacit assumption in decoherence accounts that every observer observes the same thing which Zureck does not assume and proves it.

To be exact in ignorance ensemble it is assumed its a proper mixed state so its actually in the state that's observed - since it really is in that state all observers must agree on it.

Thanks
Bill

 

[bhobba]

without quantum darwinism, you would see a cat sleeping and bhobba would see the same cat that is jumping up and down? can you give other examples for macroscopic object if quantum darwinism didnt exist.. like would you literally see window that is square while bhobba would see a triangle window? or is it all valid only for quantum object in coherence? but bhobba mentioned a cat...

That's incorrect.

First Zureck proves that observations of fragments leads to exactly the same result for all observers. There is no interpretive assumption in doing that. Secondly most interpretations like ignorance ensemble, Copenhagen etc etc make that assumption anyway because they assume a classical world observations appear in. Its an assumption of classicality all observers observe the same thing.

Zurek specifically states it:
'Bohr bypassed conflict of (i) and (ii) with (iv) by insisting that apparatus is classical, so unitarity and the principle of superposition need not apply to measurements. But this is an excuse, not an explanation. We are dealing with a quantum environment, and redundancy of previous section strengthened motivation for postulate (iii) – repeatability. Let us see where this demand takes us in a purely quantum setting of postulates (i), (ii), and (iii)'

There is no inconsistency in Copenhagen. It simply assumes something Zurek proves.

Thanks
Bill

 

[lucas_]

I didn't say that.

I said in my interpretation its not an issue because I put collapse immediately after decoherence and everything is classical after that. And I specifically pointed out Zureck has a different view and you have to read his paper to understand it rather than the limited account in Schlosshauer.

Again, as I said, there is a tacit assumption in decoherence accounts that every observer observes the same thing which Zureck does not assume and proves it.

To be exact in ignorance ensemble it is assumed its a proper mixed state so its actually in the state that's observed - since it really is in that state all observers must agree on it.

Thanks
Bill

In other words, Zurek quantum Darwinism has made an extra prediction.. in that if you can suppress quantum Darwinism, you can change the macroscopic object and make each window change shape everytime a new tourist interact with it? Do you agree with this?

 

[bhobba]

In other words, Zurek quantum Darwinism has made an extra prediction.. in that if you can suppress quantum Darwinism, you can change the macroscopic object and make each window change shape everytime a new tourist interact with it? Do you agree with this?

No.

Its simple - you are making something complex and opaque from something simple.

Most interpretations have an explicit or implicit hidden assumption that all observers will see the same thing. Such does not invalidate QM or create the issues you mentioned - in fact it specifically denies them. All Zureck does is simply prove, from the axioms of QM he states, that must be the case. It doesn't in anyway have issues with any other interpretation.

My post simply had to do with some comments he made about his axioms and his second two are somehow in conflict with the first two. That's not logically possible since the second two in fact determine the first two. But thinking about it I agree with Atty - he is being a bit loose here because he wants to show QM basically just needs his two axioms. Unfortunately Wigners theorem can't be used at this stage to get axiom 2 because probabilities haven't been introduced. Its a nice idea, and may even be true, but its not the only way to proceed eg it can be done from from just one axiom (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Which way is better? That's hard to say. I think from an understanding point of view Zureck is better because it elucidates better exactly what an observation is. The other approach is along the lines of Fuchs Quantum Information:
http://arxiv.org/abs/quant-ph/0205039

There - as in the link I gave - observations are the primitive - for Zureck states are the primitive. I personally side with Fuchs - but others may not see it that way.

Thanks
Bill

 

[lucas_]

No.

Its simple - you are making something complex and opaque from something simple.

Most interpretations have an explicit or implicit hidden assumption that all observers will see the same thing. Such does not invalidate QM or create the issues you mentioned - in fact it specifically denies them. All Zureck does is simply prove, from the axioms of QM he states, that must be the case. It doesn't in anyway have issues with any other interpretation.

My post simply had to do with some comments he made about his axioms and his second two are somehow in conflict with the first two. That's not logically possible since the second two in fact determine the first two. But thinking about it I agree with Atty - he is being a bit loose here because he wants to show QM basically just needs his two axioms. Unfortunately Wigners theorem can't be used at this stage to get axiom 2 because probabilities haven't been introduced. Its a nice idea, and may even be true, but its not the only way to proceed eg it can be done from from just one axiom (see post 137):
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

Which way is better? That's hard to say. I think from an understanding point of view Zureck is better because it elucidates better exactly what an observation is. The other approach is along the lines of Fuchs Quantum Information:
http://arxiv.org/abs/quant-ph/0205039

There - as in the link I gave - observations are the primitive - for Zureck states are the primitive. I personally side with Fuchs - but others may not see it that way.

Thanks
Bill

Has it occurred to you that quantum Darwinism is redundant? For example.. we know the window is decohered by thermal bath into eigenstates of positions, so the shape is fixed. Therefore even without different observers intercepting the photons, they would still see the same thing even by directly interacting with the window. It won't re-prepare the window because it's already decohered, so what's the purpose of quantum Darwinism, it's redundant, do you get my what I'm saying?

 

[bhobba]

It won't re-prepare the window because it's already decohered, so what's the purpose of quantum Darwinism, it's redundant, do you get my what I'm saying?

I believe its redundant by simply saying its in a proper mixed state. If that what you are saying then sure.

But you can't use that to derive Born's rule which is a biggie for Zurek. Some done think his derivation is incorrect but I don't really want to get into that - its a whole new thread.

Thanks
Bill

 

[lucas_]

I believe its redundant by simply saying its in a proper mixed state. If that what you are saying then sure.

But you can't use that to derive Born's rule which is a biggie for Zurek. Some done think his derivation is incorrect but I don't really want to get into that - its a whole new thread.

Thanks
Bill

MWI, Bohmian treat it as in proper mixed state and so is Hansen ignorant ensemble so can you confirm here quantum Darwinism is superfluous (redundant)?
What do you make of kashner partioning or factoring critique of classical pointer states. You think the reasoning is valid or just an attempt to push at her CI? Zurek just ignored it totally. I spent a lot of time reading her work yesterday.

 

[bhobba]

MWI, Bohmian treat it as in proper mixed state and so is Hansen ignorant ensemble so can you confirm here quantum Darwinism is superfluous (redundant)?

Don't know Hansen ignorant ensemble - but I can confirm the ignorant ensemble ensemble does not require a proof that observations of environment fragments give the same result.

What do you make of kashner partioning or factoring critique of classical pointer states. You think the reasoning is valid or just an attempt to push at her CI? Zurek just ignored it totally. I spent a lot of time reading her work yesterday.

Don't know that one.

But the factoring issue has been hashed out ad-nausium in many threads. It completely exacerbates me - it's the last gasp of those that won't accept progress IMHO. Its a tacit assumption of many areas of physics. Still proofs are required that show regardless of how a system is factored you get the same result. I don't want to rehash it though - its all there in the previous threads.

Thanks
Bill

 

[naima]

In this paper the author writes that the no cloning theorem is not troublesome. That the cloned things are the observables not the states. Have you the beginning of an idea on what he wants to say?

 

[bhobba]

In this paper the author writes that the no cloning theorem is not troublesome. That the cloned things are the observables not the states. Have you the beginning of an idea on what he wants to say?

You observe a system in a certain state with an observable. In decoherence the environment acts like the observable and then you observe different fragments of the environment.

Thanks
Bill

 

[lucas_]

Bill, so you basically agree that quantum Darwinism is attempt to turn improper mixed state to proper mixed state? or is it just all about explaining why pointer states are classical? And if something can explain why pointer states are classical, does it automatically explain how improper turn to proper mixed state?

 

[bhobba]

Bill, so you basically agree that quantum Darwinism is attempt to turn improper mixed state to proper mixed state? or is it just all about explaining why pointer states are classical? And if something can explain why pointer states are classical, does it automatically explain how improper turn to proper mixed state?

Quantum Darwinism is simply Zurek's attempt to reduce QM to the first two axioms he talks about in his paper. Since that doesn't include the concept of observation he has to give a fully quantum account of it, which he does via his idea of observing fragments.

Other approaches such as mine start from a different primitive - observations themselves. A tacit assumption of such an approach, even though its not explicitly stated, is that observational outcomes are objective ie everyone agrees on what it is eg if a number of people look at a readout everyone agrees its the same thing. This is really the assumption of classicality that observations are assumed to appear in.

It's two different approaches with two different primitives - in Zureck the quantum state is the primitive and observations defined from that. In the other observation leads to states via Gleason.

Take your pick - I am with observations being the primitive. But the choice isn't science - just like interpretations its simply what appeals better.

Thanks
Bill

 

[atyy]

In this paper the author writes that the no cloning theorem is not troublesome. That the cloned things are the observables not the states. Have you the beginning of an idea on what he wants to say?

I don't know what he's saying. I think it's easiest to use Nielsen and Chuang's explanation that the no cloning theorem doesn't apply if the states are orthogonal. Zurek makes the same point in a different way in http://arxiv.org/abs/1412.5206, when he talks about dividing by zero in the comments after Eq 2.

 

[lucas_]

Quantum Darwinism is simply Zurek's attempt to reduce QM to the first two axioms he talks about in his paper. Since that doesn't include the concept of observation he has to give a fully quantum account of it, which he does via his idea of observing fragments.

Other approaches such as mine start from a different primitive - observations themselves. A tacit assumption of such an approach, even though its not explicitly stated, is that observational outcomes are objective ie everyone agrees on what it is eg if a number of people look at a readout everyone agrees its the same thing. This is really the assumption of classicality that observations are assumed to appear in.

It's two different approaches with two different primitives - in Zureck the quantum state is the primitive and observations defined from that. In the other observation leads to states via Gleason.

Take your pick - I am with observations being the primitive. But the choice isn't science - just like interpretations its simply what appeals better.

Thanks
Bill

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). Is this correct?

 

[bhobba]

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). Is this correct?

This thread wasn't to discuss if Quantum Darwinism is correct - or exactly how it justifies its aim of deriving QM from the first two axioms he uses - it was to examine the issue with the axioms I alluded to.

If you want to go down that path I really think a new thread is needed.

Thanks
Bill

 

[lucas_]

This thread wasn't to discuss if Quantum Darwinism is correct - or exactly how it justifies its aim of deriving QM from the first two axioms he uses - it was to examine the issue with the axioms I alluded to.

If you want to go down that path I really think a new thread is needed.

Thanks
Bill

Ok. I was reading those axioms in the paper to understand what you and atyy were talking about when I came across this "Consensus between records deposited in fragments of E looks like "collapse". In this sense we have accounted for postulate (iv) using only very quantum postulates (i)-(iii)".

Anyway. I'd would analyze more what you and atyy were talking about. If others here know what they are talking about. Please rephrase them for some of us to comprehend. Thanks.

 

[naima]

I am beginnig to read This paper.[PLAIN]http://arxiv.org/abs/quant-ph/0307229[/PLAIN]
http://arxiv.org/abs/quant-ph/0307229
Zurek seems to be more explicit.

 

[lucas_]

He's just stating things very loosely. He can state it more properly, but he knows and his audience knows that the problems don't go away, so he sticks to the naive form of the postulates in the Copenhagen interpretation for simplicity. Because of the Born rule, in order to describe successive measurements in the Schroedinger picture, the state evolution must be random after after the Born rule has been applied when a measurement is made. Since there are two rules of time evolution - one deterministic and one random, in order to avoid a contradiction, they must be applied at different times. An external observer is needed to decide when a definite outcome is obtained, which is the measurement problem.

atyy.. I've been scratching my head for many days over this message of yours and bhobba original message. What were you two talking about? You said zurek just stating things very loosely. What must he say if he state things more tightly? And what problem he knows won't go away? Is it just the simply collapse postulate which all know pure state collapse? And what's it connection to Gleason theorem? And why did bhobba scratching his head about it initially? I'm still puzzled now over his initial puzzle. Please rephrase the issues of this not commonly known topics (many others may not even understand what you two are saying). Thanks.

 

[bhobba]

What must he say if he state things more tightly?

He would say what I said - his axiom 4 implies axiom 1 and 2 - I gave a link where I rigorously proved it using the modern form of Gleason's theorem. Hence they can't be at odds with it. Axiom 3 is implied by continuity.

He is being loose like a lot of discussion about this stuff is loose.

What he wants to do is derive 3 and 4 from 1 and 2. Part of his reasoning for wanting to do that is he thinks QM implies a collapse postulate (it doesn't) hence is at odd with axiom 2. That's false eg BM or MW does not have collapse and fully conforms to his axioms. Also even in interpretations that have collapse its often benign because a state is simply subjective - they would be interpretations such as most forms of Copenhagen. The ensemble interpretation doesn't have collapse because it's compatible with interpretations like BM that doesn't or interpretations like GRW that explicitly do in a very real sense.

That said this looseness is not confined to Zurek - its very very common. I have no doubt if you sat down with Zurek and went through it with him he would agree. His real intent into give a fully quantum account of observations.

Thanks
Bill

 

[lucas_]

He would say what I said - his axiom 4 implies axiom 1 and 2 - I gave a link where I rigorously proved it using the modern form of Gleason's theorem. Hence they can't be at odds with it. Axiom 3 is implied by continuity.

He is being loose like a lot of discussion about this stuff is loose.

What he wants to do is derive 3 and 4 from 1 and 2. Part of his reasoning for wanting to do that is he thinks QM implies a collapse postulate (it doesn't) hence is at odd with axiom 2. That's false eg BM or MW does not have collapse and fully conforms to his axioms. Also even in interpretations that have collapse its often benign because a state is simply subjective - they would be interpretations such as most forms of Copenhagen. The ensemble interpretation doesn't have collapse because it's compatible with interpretations like BM that doesn't or interpretations like GRW that explicitly do in a very real sense.

That said this looseness is not confined to Zurek - its very very common. I have no doubt if you sat down with Zurek and went through it with him he would agree. His real intent into give a fully quantum account of observations.

Thanks
Bill

Is Gleason's theorem completely compatible with MWI, Bohmian, Zurek, Ignorance essemble, GRW, etc? what interpretation is it not compatible with?

 

[atyy]

Part of his reasoning for wanting to do that is he thinks QM implies a collapse postulate (it doesn't) hence is at odd with axiom 2. That's false eg BM or MW does not have collapse and fully conforms to his axioms. Also even in interpretations that have collapse its often benign because a state is simply subjective - they would be interpretations such as most forms of Copenhagen. The ensemble interpretation doesn't have collapse because it's compatible with interpretations like BM that doesn't or interpretations like GRW that explicitly do in a very real sense.

It's a matter of terminology. One can consider standard QM to have collapse because
(1) it is unknown whether there is a BM or GRW for all domains of QM, including the standard model
(2) even if all domains of standard QM can be embedded into BM or GRW, because these interpretations intend to solve the measurement problem, and at that level they must deviate from QM, and so are not strictly equivalent to QM
(3) many-worlds, if it works, is presumably exactly equivalent to QM, but there is still no consensus on whether it works, even by proponents such as Wallace and Carroll

So the orthodox Copenhagen-style interpretation with collapse is the only version of QM in which there is consensus that it works in all domains of QM. It is true that some consider collapse to be not a problem at all, not even pointing to an incompleteness (eg. Griffiths's version of consistent histories). However, by and large most orthodox Copenhagen-style interpreters do agree that Copenhagen does have a measurement problem, including Dirac, Landau and Lifshitz, Weinberg, Bell and Tsirelson. (Bell is often not considered a Copenhagen-style interpreter, but I included him here because he acknowledged it works FAPP, and used it to derive that QM violates his inequality.)

 

[bhobba]

Is Gleason's theorem completely compatible with MWI, Bohmian, Zurek, Ignorance essemble, GRW, etc? what interpretation is it not compatible with?

None.

Thanks
Bill

 

[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?