# Paper on Bell Locality

1. Feb 2, 2006

### ttn

I've argued here in the past (with dr chinese and others) about what, exactly, is proved by Bell's Theorem. Here is a new paper which addresses
and clarifies many of those points:

http://www.arxiv.org/abs/quant-ph/0601205

I suspect it will be of interest to people here. But before Patrick jumps on me about MWI, I'll just say this: I am taking it as given that the measurement in each wing of these EPR/Bell correlation experiements has a definite outcome. Given that reasonable assumption, the conclusions in the paper follow. But if one holds (with MWI) that these experiments do not have definite outcomes, all bets are off.

2. Feb 2, 2006

### vanesch

Staff Emeritus

Hehe ! This brainwashing has had its effect all right
I'm staying out of here :!!)

3. Feb 3, 2006

### Sherlock

Thanks for posting the link to your "Bell Locality and the Nonlocal Character of Nature", ttn. Ever since you last visited this forum I've been meaning to reread your "EPR and Bell Locality", and research some other papers in order to better understand the argument(s) you advocate. Your latest paper looks like it might make that task a bit easier in a way.

Vanesch has been kind enough in a recent thread to discuss with me at some length his thinking behind his adoption of the MWI. But I still think that the reasoning behind the idea that experiments have definite outcomes is much stronger than the reasoning behind the idea that they don't.

4. Feb 3, 2006

### vanesch

Staff Emeritus
It's damn hard to stay out of here :grumpy:

The statement by the OP illustrates my POV: you'll have to choose between some form of non-locality and parallel outcomes if the empirical predictions of QM are correct.

Given the disaster of the first option, and the fact that the second option is already foreseen in the formalism of QM, I go for the latter: it is MWI or non-locality, in a nutshell.

Last edited: Feb 3, 2006
5. Feb 3, 2006

### Tez

The nut in this shell prefers to go for nonlocality rather than an excess of ontological baggage

6. Feb 3, 2006

### vanesch

Staff Emeritus
Ok. Bye, general relativity, then...

7. Feb 3, 2006

### Tez

I'll worry about that if and when we have a quantum theory of gravity. At the current energy scales there's no particular conflict, and I certainly lean to a belief that quantum mechanics reveals more of the "reality of the universe" than GR. However I highly doubt it'll be a problem, for the simple reason that I doubt by then we'll be seeing space and time as having the ontological status that GR gives them...

8. Feb 3, 2006

### ttn

I agree with Tez, but I really don't think there's any point arguing about this. Neither side will convince the other.

Perhaps a more interesting question for discussion would be what *other* possible routes exist to avoid the conclusions in the above paper. Sure, you can get around the conclusion if you deny that experiments actually have definite outcomes. But if someone wasn't comfortable denying that (and many people, I think rightly, aren't), what *else* could they deny instead if they wanted to avoid the conclusion that nature is non-local?

There is a long history of proposed answers here, e.g., people saying you can have a local theory as long as it doesn't have any hidden variables, or you can have a local theory as long as it isn't deterministic, etc. Does anyone think those positions are viable? Does anyone think there is some other principle that can be rejected instead of locality (I mean other than the principle that when we see a pointer pointing left it's actually pointing left)?

9. Feb 3, 2006

### Tez

Well, theres this somewhat strange "blockworld" idea, based on the observation that one has to eventually "bring the notebooks together and compare data", that the manifestation of Bell correlations is due to the events in some sense having a common future rather than a common past.

But I don't claim to really understand it, or the the appeal of it, since it then seems to deny my free will not to meet up with the other person, and once one starts allowing such conspiracies its easy to get pretty much whatever you want via the backwards lightcone....

10. Feb 3, 2006

### DrChinese

I am looking at the paper. Should be the seed for some good discussion in there.

ttn, I had linked your (?) earlier paper ("EPR and Bell Locality") on my site a while back because I liked some of the things it covered. Even if I don't always agree...

In the meantime, Tez* has taught me another lesson in why it is always good to stay close to the formalism. So I will stick with the "either/or" for now.

(*Thanks again Tez; I am still digesting steering theorems but suspect this may be beyond my reach; but I haven't given up yet.)

11. Feb 3, 2006

### vanesch

Staff Emeritus

The main argument for MWI is not locality of course, it is unitarity and the refusal to give special "ad hoc" status to the physics of measurement apparatus (you cannot, in the standard view, DERIVE the hermitian "measurement observable" of a device, if the quantum physical description of the device is given ; you have to decide "ad hoc" what is its measurement basis). Decoherence can give you the measurement basis, by looking at the pointer states which are stable against interactions of the environment. But decoherence only makes sense in an MWI setting. You get the locality (and the resolution of the apparent non-locality and spooky action at a distance in EPR situations) for free.

MWI is based upon taking unitarity seriously. And then you get locality for free, if the unitary interactions of the theory are local (which they are).

12. Feb 3, 2006

### vanesch

Staff Emeritus
I think that an explanation that is possible is that nature is inconsistent (in the sense that it cannot be described by a mathematical structure). Before crying wolf, this is exactly Bohr's point of view. Quantum theory does not describe nature, it just gives you relations between experimental preparations and outcomes. And when you DO try to describe nature (using hidden, or not, variables) you run into problems. So microphysics is NOT DESCRIBABLE by a mathematical structure. Maybe it even doesn't exist, and measurements "just are". Or "knowledge just is". And quantum theory is the statistics of inconsistency.

Another explanation that is possible, is solipsism. Quantum theory describes correctly our sensations, but there's no underlying reality that is responsible for it (and hence we will not find any mathematical structure that can do so).

Seriously, when I look at the alternatives (non-locality, inconsistency, solipsism), I think that MWI is not so bad!

EDIT: I forgot of course the most prozaic alternative: EPR situations do not exist, and the empirical evidence for it is flawed (the local realist loopholes).

Last edited: Feb 3, 2006
13. Feb 4, 2006

### Tez

Most Everettians I've talked to say that the things with ontological status in their view (quantum states) are nonlocal. (Sure, when doing a Bell experiment the Hamiltonians are local, but that doesn't make the whole description local - give me nonlocal Hamiltonians and I'll violate a Bell inequality for you with an unentangled "local" wavefunction!)

I think you have to go further than saying MWI is based upon taking Unitarity seriously. IMO it is based on taking one particular mathematical formulation of QM seriously, and in particular the tensor product structure is simply presumed (I'm far more worried about "preferred tensor product structures" than "preferred bases", since the former speaks to our notions of systems and locality far more deeply).

Let me elaborate on this "one particular math formalism point". I'm going to cut and paste from an email exchange, sorry!
>
> Imagine that we lived in a universe in which the Wigner distribution
> for every system was positive. That is, all of standard quantum
> mechanics is true (Born rule etc), but an additional restriction
> forces positivity of every Wigner distribution. The set of states with
> positive Wigner distribution is very large - it includes coherent
> states, and entangled states - such as squeezed states, in fact it
> includes all Gaussian states, but many other states as well. They form
> a convex set and thus a completely self contained subset of quantum
> mechanics (e.g. All Hamiltonians would be the same - implying the same
> hydrogen atom spectrum - and one can't evolve out of this set given
> only states within it). Interestingly, since the original EPR argument
> used only the
> (position-momentum) squeezed state and Gaussian type measurements,
> they could also have had an EPR paper and Bohrian answer!
>
> Now lets also imagine the people in this universe do not actually have
> the Wigner formalism - they only have the standard Hilbert space
> formalism. Thus they are writing down states in a Hilbert space,
> describing their measurements in the standard von-Neumann way as
> giving an entanglement between the system and apparatus and so on.
> They see an "intrinsically" probabilistic theory and nonseparability.
> They go through the same metaphysical convulsions we do about the
> collapse of the wavefunction.
>
> If these people adopt an Everettian approach to understanding the
> underlying reality implied by their physical theory, is it really
> justified? If one advocates that Everett follows from just accepting
> the math of QM, then it should be just as applicable to this universe
> - the math is identical. Only the original set of states is different.
>
> But in such a universe it is clear - the physicists have been tricked
> by their mathematics (inseparability of states with respect to a
> particular tensor product, a belief in "objective" probabilities and
> so on). In fact there is this perfectly good realist explanation (in
> terms of the Wiger probability densities as classical uncertainties
> over a phase
> space) lurking out there.
>
> Until I am convinced that this is not a good analogy for where we're
> at with quantum mechanics as it stands, I'm not prepared to take what
> I see as an extremist way out!
>

My point in that email is that the eye-glazing wonder an Everettian feels when they see
(|0>+|1>)|0>
evolve to
(|0>|0>+|1>|1>)
would be felt by inhabitants of "Gaussian World", since they have an identical Hilbert space structure. They may well adopt MWI. The only difference between that world and ours, IMO, is that we haven't found the equivalent probabilistic description over a suitable "ontic phase space".

I believe your other argument about not being able to "derive" the measurement observable of the device given its physical description as a particular justification for MWI is spurious. In Liouville mechanics, if I give you the Liouville distribution of one system (the apparatus) and the interaction Hamiltonian between it and some other system (which is described by another Liouville distribution), and you then evolve them both to the coupled (joint) Liouville distribution, you cannot "derive" what observable it corresponds to either. In this purely classical situation, just as in the quantum case, some other physical (generally empiricist) input is required.

I'm off to Pareee until monday afternoon, I look forward to reading your reply when I get back.

Tez

14. Feb 4, 2006

### vanesch

Staff Emeritus
I find this a rather amazing statement ; I didn't know that the positivity of the Wigner function (which turns it into a genuine probability densty in phase space) was conserved under unitary evolution!

http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution

But if what you write is true then I don't see how one could ever arrive at a Bell theorem violating state. After all, the x and p can serve as the "hidden variables" in this case (with distribution given by the Wigner function).

I'm even wondering how one could avoid obtaining non-coherent states from unitary evolution if we start out with only coherent states. After all, isn't that exactly what happens in, for instance, a PDC xtal ? A coherent pump beam is directed onto the system, which evolves into a non-coherent state of two entangled beams.

I understand your POV, I think. In fact, in standard classical probability theory, there is equivalence between taking several, parallel universes with weights given by the distribution over phase space, and saying that this is overkill and that there is ONE ontological universe, and that the others have never existed, and were simply part of our ignorance. So it would seem justified, in this view, to eliminate the parallel universes, because they could be eliminated all along. Nevertheless, they could be considered too. There's nothing WRONG with considering parallel worlds when doing classical probability. Only, there's no compelling reason to do so. But it is not wrong.

However, I would think that Bell's theorem and its violation indicate us that we WILL NOT FIND such an underlying distribution (if I understand things correctly, a positive Wigner distribution would allow exactly that: have a pre-existing phase space distribution of hidden variables which "tag" the individual parallel worlds with a definite probability) - at least, if we're bound to have only local dynamics on this phase space - Bohm's theory being an example of the possibility of doing so if this condition is relaxed.

As to the factorization of Hilbert space, you're right that it is the choice of factorisation which determines the basis and everything. But I think that there is a very evident factorisation: H_observerbody x H_rest
After all, what we need to explain is how the observer (connected to a body, that is to say, a certain number of physical degrees of freedom) is to see the rest of the universe.

Probably more later, have to go...

15. Feb 4, 2006

### vanesch

Staff Emeritus
I don't understand this: after all, we will get out a distribution of probabilities of the state of the apparatus (integrating over the distribution of probabilities of the other system). That's good enough: you will find a probability distribution for the readings of the display of the instrument (5% chance that it reads "3 V", 20% chance that it reads "8 V" etc...). That is because each individual state of the apparatus (each point of its part of phase space) corresponds to a specific display reading.

But that is NOT the case for a quantum apparatus, which can be in superpositions of "classical display reading states". We can have just as well the state |3V + 8V> = |3V> + |8V> as the state |3V-8V>. But we don't know what to do with these states. So IN ORDER FOR US to be able to say that we should calculate probabilities in the {|3V>, |8V>} basis, we have already to say that this is the relevant basis, and not the |3V+8V> and the |3V-8V> basis. We have to pick "pointer states", which correspond to classical states. Once we have defined these pointer states, we can trace back through the unitary dynamics of the apparatus, and see with what system states this corresponds, and once we've done that, we know what are the "eigenspaces" on which the hermitean observable is to be constructed.
For instance, when having an instrument with a dial, one has to choose POSITION STATES of the dial as pointer states, and not momentum states of the dial, or others. If we were to choose wrongly, the momentum states, we would find that the instrument measures an entirely different quantity than we thought (for instance, if the instrument was thought to measure a particle momentum, then suddenly, it will turn out to measure a particle position). If we were going to apply the Born rule in this "wrong" pointer basis, we would come to entirely different conclusions about the behaviour of the microsystem.

In the classical view, ALL instrument states are classical states, hence we have not this difficulty. Of course we cannot really know what the apparatus is supposed to measure until we've said WHAT aspect of the state of the apparatus is the "measured quantity" (in our case, the display - I suppose this is what you are alluding to), but this has no incidence on the calculated probability distribution.

16. Feb 7, 2006

### Tez

A squeezed state (the output of a PDC) is still a gaussian state!

I'm not quite sure where your confusion is. Is it my claim that they could still have entanglement and an EPR paradox even though there is a hidden variable model given by the Wigner function? Check out chapter 21 of Speakable and Speakable, Bell discusses it very briefly there.

Of course, even though the EPR state has a positive Wigner distribution it could be used to demonstrate nonlocality, but only if you go beyond things like position and momentum measurements, which are all Gaussian. Bohr's reply to EPR also only used gaussian state arguments and so would remain equally valid.

It is a mathematical fact, but one that I cant think of how to prove to you in a few lines on an internet forum, that if everything starts off gaussian then nothing you can do (under energy conserving Hamiltonians) will take you out of such. I'll hunt around for some notes on this and try and stick them on my webpage...

More later,
Tez

17. Feb 7, 2006

### vanesch

Staff Emeritus
Ok, I did a small calculation myself.
I started with the entangled state:
$$-e^{-{\left( -5 + x1 \right) }^2 - {\left( -6 + x2 \right) }^2} + e^{-{\left( -1 + x1 \right) }^2 - {\left( -2 + x2 \right) }^2}$$

Clearly, these are two entangled gaussian states:
|1>|2>-|5>|6>

where |u> stands for exp(-(x-u)^2)

These are entangled gaussian states (even with average momentum 0!).

When I apply, using mathematica, the formula for the Wigner distribution, after some crunching of the command:

I obtain:
$$\frac{e^{\frac{-8\,\imag \,{p1} - {{p1}}^2 - 8\,\imag \,{p2} - {{p2}}^2 - 4\,\left( 61 - 2\,{x1} + {{x1}}^2 - 4\,{x2} + {{x2}}^2 \right) }{2}}\, \left( e^ {4\,\imag \,\left( -28\,\imag + {p1} + {p2} \right) } - e^{8\,\left( 9 + {x1} + {x2} \right) } - e^{8\,\left( 9 + \imag \,{p1} + \imag \,{p2} + {x1} + {x2} \right) } + e^{4\,\left( \imag \,{p1} + \imag \,{p2} + 4\,\left( {x1} + {x2} \right) \right) } \right) }{2\, {hbar}}$$

And when you plot that function, say, for p1 = p2 = 0, then you have two POSITIVE bumps, around {1,2} and around {5,6} (as expected), but ALSO A NEGATIVE BUMP around {3,4}.

So I still don't understand your statement.

18. Feb 7, 2006

### vanesch

Staff Emeritus
Ok, another, slightly more realist situation:

Take the function of entangled gaussians:
$$-e^{-{\left( -5 + {x1} \right) }^2 + 5\, i \,{x1} - {\left( -6 + {x2} \right) }^2 + 3\, i \,{x2}} + e^{-{\left( -1 + {x1} \right) }^2 + 3\, i \,{x1} - {\left( -2 + {x2} \right) }^2 + 5\, i \,{x2}}$$

It is the same as before, except now that we have genuine position and momentum entanglement:
|1> has a central momentum of 3 and |2> has a central momentum of 5, while |5> has a central momentum of 5 and |6> has a central momentum of 3.

We can again do the same computation, and now the wigner function is:
$$\frac{e^{-139 + 3\,{p1} - \frac{{{p1}}^2}{2} + 3\,{p2} - \frac{{{p2}}^2}{2} + 4\,{x1} - 2\,{{x1}}^2 + 8\,{x2} - 2\,{{x2}}^2}\, \left( e^{2\,\left( 56 + {p2} \right) } + e^{2\,\left( {p1} + 8\,\left( {x1} + {x2} \right) \right) } - 2\,e^{73 + {p1} + {p2} + 8\,{x1} + 8\,{x2}}\, \cos (2\,\left( -16 + 2\,{p1} + 2\,{p2} - {x1} + {x2} \right) ) \right) }{2\, {hbar}}$$

if I didn't make any mistake. For x1 = x2 = 4 and p1 = p2 = 4, it shows again a negative bump.

EDIT: I noticed that there is a problem displaying the imaginary unit I with the TeX display. Tried to fix it, but it didn't work...

EDIT2: ah, it worked now... (probably a cache problem)

Last edited: Feb 7, 2006
19. Feb 7, 2006

### slyboy

A superposition of gaussian states is not itself a gaussian state.

20. Feb 8, 2006

### Tez

Hi Patrick

As slyboy said - a superposition of gaussian states is not a gaussian state (the convexity I referred to above is under mixing, not coherent superposition).

But I realised while re-reading my notes from when I was thinking about this that the terminology I was using ("conservative Hamiltonians"), is a restricted class of Hamiltonians, and in my idiocy I wrote "energy conserving Hamiltonians". To get your superposition example give would require a Hamiltonian outside of this class. Sorry for wasting your time with that. [The stupid terminology comes from the analogy with classical mechanics where conservative Hamiltonians preserve the area of a Liouville distribution - likewise these Hamiltonians preserve the areas of the the Wigner distribution, thus keeping minimal uncertainty (gaussian) states as minimal uncertainty states].

The conservative Hamiltonians, which I'll simply define as those that preserve the gaussian nature of states, include basically everything representable as a quadrtic form in the canonical variables. Examples include the standard harmonic oscillator, the squeezing Hamiltonian you alluded to above (which will create the entangled states from unentangled ones), and, more interestingly for this particular discussion, the "x*p" Hamiltonian. This latter is the sort of Hamiltonian commonly invoked in a von-Neumannesque description of a position measurement: We start with a delocalised particle (e.g. in a momentum eigenstate, which is Gaussian) and a massive "pointer" system in a gaussian state well localized in position, and then couple the two systems under the xp Hamiltonian. Thus "gaussian world" has a "measurement problem". As I mentioned above, it also has all the same states and Hamiltonians as required to perform all elements of the EPR/Bohr argument (which despite a myriad of papers claiming the contrary clearly has nothing interesting to say about locality, it is an argument about the completeness of QM).

So the guassian toy universe has a standard Hilbert space representation which its inhabitants may well be tempted to try and understand using a MWI. However, once someone in this universe found this concrete Wigner distibution "hidden variable model", I'd imagine that those musings would be dropped quite quickly and there'd be a lot of embarrassed physicists looking around wondering why they'd ever entertained such beliefs. (I am not saying this to be insulting or polemic, I genuinely think something similar will happen with full QM.)

Some other comments based on your various postings: As you noted Bell's theorem tells us nothing about whether or not there exists a classical probability type of interpretation of quatum mechanics, only that whatever the underlying "ontic states" are (the equivalent of the position/momentum phases space for gaussian world say) they must be nonlocal. Is this so bad? As I mentioned above I have never heard an MWI'er claim that they have local ontic states, I'd be interested if this is your claim.

Regarding the "preferred factorizations" of the Hilbert space, I think there are a myriad of interesting problems to be tackled, but this isnt the place to list them I guess. [A new example that just popped to mind and thus may well be trivial: When one moves from non-relativistic QM to QFT, certain factorizations can get "mixed up". An example is spin and momentum, which non-relativistically are described in a tensor product, but which by relativisitic observers are described in a direct sum. These two possibilities yield quite different ontologies under MWI, as I understand it] I do agree that the "observer/rest" split is about as natural as one can get. However my (perhaps limited) understanding of standard MWI is that actual splittings are not limited to observer/everything else situations, and thus an ambiguous ontology is somewhat inevitable.

Regarding the Liouville distribution stuff: The reason you can claim that in the classical case one can simply look at the marginal of tha appratus system is because you and I know that the relevant phase space variables we want to describe the world in are position and momentum. However I can take any old canonical re-definition of the phase space variables (x->x+p,p->x-p or something), and give you the Liouville distribution in terms of these variables. My claim is simply that some form of empirical input is required in order to get the "correct" interpretation of the measurement. In the quantum case, (under the view the QM is incomplete) we simply cannot expect to extract the information you desire. Let me harp back to gaussian world (which, although you may not agree with how I'm using it I hope you see is a useful pedagogical tool!) - your argument applies with no modification to it. There, however, its clear that what you are expecting of the theory is something one can be tricked into by an assumption of completeness of the particular Hilbert space formalism.

Oh, another thing popped into mind: A different sort of problem with regarding quantum states as ontic entities arises from the fact that in many physical situations I can use completely different quantum states to describe exactly the same physical situation. It can be done in a way that there is no operational way to distinguish which description is "correct". An example is the case of the laser, where some people advocate assigning a certain mixed state while others a pure coherent state, but there is provably no way to distinguish the two. [I have a light reading paper http://www.arxiv.org/abs/quant-ph/0507214 if you want to see how I think these situations should be understood, I am the middle author] From the point of view that quantum states are an incomplete encapsulation of an observer's information about the world, this presents little problems. But I suspect an Everettian must insist that there is once "actual fact" about which state is the correct description. Since there is no way to determine whether they are correct, I further suspect there is a dangerous philosophical cliff just waiting for them to fall off :) . Its not a well thought through argument, don't respond unless you think its interesting!

Man, this is one of the longest posts I've ever made on a forum. If one of my students catches it they'll be moaning all day at me! Travis I'm sorry for the derailment - if a moderator wants to split this discussion off somehow that'd be fine.

Finally let me point out that I am agnostic regarding interpretations, though I think the evidence suggests that none of them are completely satisfactory, but that the quantum state is incomplete. This latter assertion is not an interpretation because we have no concrete example of how to quantify this intuition and represent quantum states as states of knowledge over some sort of underlying ontological state space. If I was a betting man I'd bet its possible though.

Tez

Last edited: Feb 8, 2006