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    Quantum Casino - Less Than Zero Chance
    By Johannes Koelman | October 24th 2012 11:34 AM | 36 comments | Print | E-mail | Track Comments
    About Johannes

    I am a Dutchman, currently living in India. Following a PhD in theoretical physics (spin-polarized quantum systems*) I entered a Global Fortune

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    Human thought has led to a variety of remarkable and profound insights. Many of these insights are well established and have been embraced by a significant portion of the global population.

    The earth being round, the atomistic nature of matter, our unremarkable place in the universe, and us being a product of evolution, all being examples of such insights. Other insights, although unanimously embraced by experts, have a long way to go for a larger population to accept them. More than for any other subject, this holds for quantum physics. No other product of human thought is as profoundly mysterious as quantum theory.

    Unfortunately, the subject is not easy to digest and, to make things worse, it is often misrepresented in the pop-science literature. Those eager to understand the quantum are often fed with confusing slogans and misleading analogies.

    A year ago I dedicated two blogposts (here and here) to the mysteries of the quantum. Judging from the reactions, these posts must have been useful to at least a few of the readers. The simple thought experiment presented (dealing with Albert's socks) allows the reader to explore the weird world of quantum physics, an experience that likely will challenge the reader's view on reality. A view that - for all of us - is heavily biased by sensory perceptions limited to classical (non-quantum) physics.

    Yet, I feel one question remained insufficiently discussed in my earlier posts. And that is the question how physicists deal with quantum reality. Let's see if I can fill this gap by further exploring Albert's quantum socks.


    ALBERT'S SOCKS



    Albert's chest of quantum drawers, although most spooky in its behaviors, appears rather unremarkable and featureless from the outside. Three rows, each consisting of three drawers, make up this piece of furniture. Each night, when Albert is sound asleep, his housekeeper fills this cabinet with fresh socks. The next morning, whenever Albert pulls open a drawer, he is presented either with a single sock or an empty drawer. Unfortunately for Albert, he can not open just any number of drawers. Each morning, his search for fresh socks is limited to the opening of three drawers: any three drawers forming a horizontal row or any three drawers forming a vertical column.

    Initially, out of habit, each morning Albert opens a horizontal row of drawers. He does this without giving much thought to which of the three rows to open. After a few days, he starts noticing a pattern. Whichever row he opens, either all three drawers are empty, or two drawers each contain a sock leaving one empty drawer. In other words, a row of drawers always contains a total even number of socks. Many days pass by, and Albert never encounters a row not containing an even number. That makes sense: each row containing an even number of socks tells Albert that each morning the chest must be filled with an even number of socks. An observation comfortably compatible with the notion that socks come in pairs.

    One morning, Albert decides to deviate from his fixed ritual, and he opens a vertical column of drawers: the three leftmost drawers. Interestingly, this time he observes an odd number of socks distributed over the three drawers opened. The next day, he opens the same leftmost column of drawers, and again observed an odd number of socks.

    Albert gets curious about the other columns. What number of socks will they reveal? The next morning he opens the middle column. Again an odd number of socks. The next day he once more checks the middle column. Once again he observes an odd number of socks.

    Albert is a clever guy, and he now realizes he can predict with certainty that the rightmost column of drawers must behave differently from the two columns already inspected. The rightmost drawers for sure must contain a total number of socks that is even. This is obvious as an odd number of socks in the rightmost drawers, added to the number of socks in the middle column (observed to be odd) and the leftmost column (also observed to be odd) would result in an odd total number of socks in the chest. A contradiction, as he had already observed, based on opening horizontal rows, that the total number of socks in the chest is always even.

    The next morning he eagerly opens the three rightmost drawers. To his astonishment he observes an odd number of socks.

    The next few mornings he randomly checks the various columns. Each attempt reveals an odd number of socks. Albert realizes something must have gone wrong. Maybe his housekeeper coincidently changed from filling the chest with an even number of socks to filling it with an odd number of socks, just at the time he switched over from opening horizontal rows to vertical columns?The next morning Albert again opens a horizontal row. An even number of socks stares him in the face. He starts randomly switching between horizontal rows and vertical columns. Horizontal rows always deliver even numbers, vertical columns odd numbers.

    This drives Albert crazy. The results he is obtaining are logically impossible. "At any given morning if I would open three rows", Albert reasons, "I would end up with an even number of socks. However, would I open three columns, I would end up with an odd number of socks. Yet in both cases I would have opened the same nine drawers. How can this be?"


    SHUT UP AND CALCULATE!



    "Nice story" you might react, "but nothing more than a fantasy. Obviously chests of drawers like these logically can't exist in our world".

    Well, as a matter of fact, they do. I can put that even more strongly: all evidence points at our world consisting solely of devices like Albert's drawers. It just happens to be that these drawers tend to be very, very small and that we generally observe the aggregate behavior of many drawers without being able to distinguish critical details such as the distinction between even and odd counts.

    Many physicists deal with quantum reality day in, day out. How have they come to terms with a physical reality that presents us with devices like Albert's chest of drawers? Physicists don't seem to lose any sleep over the many counter-intuitive notions and apparent paradoxes surrounding quantum theory. Are they oblivious to the utterly strange world view that the quantum represents?

    Not at all. The point is that the vast majority of physicists simply have stopped worrying and have embraced a practical approach described by the catch-phrase:
    "Shut up and calculate!"
    This phrase was coined by Cornell physicist and science communicator David Mermin, in an attempt to mock the widely-accepted Copenhagen interpretation of quantum physics. Mermin was not at ease with the Copenhagen position put forward by Niels Bohr in the famous Bohr-Einstein debates. Ironically, two things happened following the publication of Mermin's phrase. Firstly, Mermin's own ideas on how to interpret quantum physics started to converge towards the Copenhagen interpretation. Secondly, the "shut up and calculate" phrase achieved the status of a popular instrumentalist approach to quantum physics equated with eschewing all interpretation.
    Let me try to offer you the "shut up and calculate" approach as a choice for coming to terms with the quantum. To do that, I need to explain how to calculate sock counts for Albert's drawers.


    LOADED DICE



    The mathematical machinery behind quantum devices like Albert's chest is compelling and carries a great beauty. It is based on a generalized probabilistics that adds likelihoods not as a linear sum, but rather as a Pythagorean sum. I would love to present this quantum math to you. But alas, the math is too involved to explain in a blog like this.

    Fortunately, the complex quantum math built on Pythagorean sums can be mapped on the much simpler classical math based on straightforward linear sums. Physicists apply such mappings all the time when studying what is known as "the classical limit" of quantum systems: the behavior of large quantum systems that can be accurately represented with classical (non-quantum) laws of physics.

    Eugene Wigner, one of the quantum pioneers, discovered in the early 1930's that something weird happens when approximating quantum physics with classical probabilities. He discovered that one can describe quantum systems with probabilities that add linearly, but these probabilities are no longer guaranteed to be non-negative.

    Paul Dirac, Wigner's brother in law, later wrote a paper that discusses the use of concepts like negative energies and negative probabilities in quantum physics:
    "Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative of money."
    Years later the idea of negative probabilities received increased attention when Richard Feynman started to popularize the idea:
    “Trying to think of negative probabilities gave me a cultural shock at first, but when I finally got easy with the concept I wrote myself a note so I wouldn’t forget my thoughts. . ."
    In describing Albert's chest of drawers, how far do we get with negative probabilities? Before you read my negative probabilities description for Albert's chest, I urge you to try it yourself first. It's a straightforward and instructive exercise.

    Ok, here is a probabilistic description of Albert's drawers. A total of 10 configurations are relevant:
     
    1) Nine configurations consist of two drawers in a single row being filled, and all other drawers being empty. These nine configurations each carry a 1/6 probability.
     
    2) One additional configuration features an empty chest of drawers. This configuration carries a minus 1/2 probability.

       

    It is easy to see that all probabilities add up to unity, as they should. The negative probability for the 'all empty' state is a strange beast, but a beast that doesn't show up in the end result when calculating observable probabilities.
     
    For instance, let's calculate the probabilities for observing the various numbers of socks in the bottom row of Albert's chest. In the above table you can read off that we have seven realizations with an empty bottom row, six with +1/6 probability, and one with -1/2 probability. The total probability for an empty bottom row is therefore +1/2. Furthermore, we have three realizations with a bottom row filled with two socks, each with a probability of +1/6. The total probability for finding two socks in the bottom row is therefore +1/2. So we have two equal likelihoods of finding zero or two socks, each with a positive (+1/2) value, and we are guaranteed to retrieve an even number of socks from the bottom row.
     
    Repeating the same calculation for a column leads to the result that a column renders one sock with unit probability, hence a guarantee for an outcome of an odd number of socks.
     
    What happened in both these cases is that the negative probability gets cancelled by positive probabilities describing the same outcome for the row or column under consideration. As a result, rows lead to even numbers and columns to odd numbers, and never does a negative probability turn up its ugly head.
     
    This obviously doesn't hold when determining the number of socks in the entire chest of drawers. A negative probability does show up prominently, and it does not get cancelled as there are no positive probabilities describing the same configuration. However, we don't need to worry about this, as the configuration with negative probability does not correspond to a possible observation. Only three drawers (one row or one column) can be opened, not the whole chest.
     
    What this means is that one can extract only a limited amount of information out of Albert's chest. Getting more information out of it (opening more drawers) is impossible as it would lead to negative probabilities to become evident. This is a manifestation of Heisenberg's uncertainty principle for Albert's chest.
     
    Heisenberg uncertainty, destructive interference, spooky action at a distance, violation of Bell's inequalities, all of quantum physics can be mapped on negative probability models. So why don't physicists use such negative probability models? The answer is simple: negative probability models are rather clumsy representations of what is better described using "Pythagorean probabilities".
     
    But as a model to get some understanding of the fundamentals of quantum physics, negative probability models do carry a value largely unexplored by pop-science writers.
     
    All of this places some of Einstein's quotes in a different light.
     
    God does play dice. And malicious he is. Or how would you characterize anyone who throws dice loaded to render negative chances, and who manages to keep this so well-hidden from us?
     

    Comments

    A bit off topic, but it occurred to me that an actual set of draws that has this observable behavior is quite easy prepare if the many worlds interpretation holds. It can be done as a straight forward application of quantum immortality. Although I don't think that route has any interesting probability math involved, hence off topic.

    Johannes Koelman
    Thanks for this remark. Reminds me that I was planning to touch upon the Many Worlds Interpretation. Simply forgot about it when writing the blogpost (in hindsight not too bad, as the article was already growing into a too large post).

    I think you are arguing that MWI can be seen as compatible with a negative probability interpretation for Albert's drawers (with the absolute values of all relevant probabilities adding up to 2)?
    No, I've no idea what the math does, my point was far cruder.
    You take a regular set of draws, hook up some sort of interlock so you can only open one row or column, and toss some socks in it. Some of the rows and columns will violate the rules, you hook those up to the trigger of a small thermonuclear device.
    Regardless of what the rest of the math does the probability of seeing row with an odd number of socks or column with an even number is now zero.
    Quantum immortality then provides for you to always live to observe the desired result.

    Johannes Koelman
    I see. You can somewhat refine this demolition model as you can always fill the chest of drawers such that no more than one drawer can trigger a wrong even/odd count. Only the opening of this specific drawer needs to cause a nuclear explosion.  (A big one, I assume, as you need to annihilate the whole universe, right? :) )
    No, you just need to make sure Einstein's chance of survival is effectively 0.
    Your description talks only of the behavior Einstein observes, with this setup (and many worlds) he can go on opening draws morning after morning indefinitely, always observing the puzzling behavior.
    Of course it looks rather different to the rest of us, we see his entire neighborhood reduced to a smoldering crater within a couple of days of starting this exercise, with an apparent 1 in 6 chance of it happening on any particular day.
    What his neighbors observe under this setup is kinda interesting. For them the experiment rolls on indefinitely... until they go on holiday. Every morning they are outside the 100% zone there is an apparent 1 in 6 chance of the bomb going off.

    vongehr
    You say that Pythagorean sums are too difficult for a blog, but they are actually straight forward to understand as how shadows (projections) obviously behave (I described this on a lay person (blog) level here, you are welcome to contribute/criticise).

    Instead, you support tossing out the positivity of probabilities while on the other hand keeping normalization, which I find immediately and obviously less helpful from all perspectives, be it arbitrariness of what is let go or interpretational issues, but especially on a blog.

    The underlying reason for refusing intuitive models (which you usually like and are quite a master at coming up with) seems to be your support of "genuine stochasticity" (God plays dice). I encourage you to step back and look at that any theory that is not already fully internally "parametrized" (i.e. still needs external "time of time" (flow of time), "random stochasticity") is not a satisfying model but a regress error (premature termination of regress at least). In a satisfying model of randomness, randomness is by definition not any longer something external to the model (God).

    I think you should focus your skills in comming up with intuitive models that advance, for example, EPR MWI models like I propose, because in those models, there is no longer external randomness/time that allows time to flow so that we drift to branching points where God then throws a coin. EPR-MW models need examples for how the number of microstates is consistent with the quantum probabilities that the models already allow. I think this is very fitting to your skill set, judging from all the models you came up with (they are usually microstate counting heavy, which is exactly what I need).
     
    Johannes Koelman
    Thanks for the compliments. Yes, we are in a strong and solid agreement on the virtues of micro state counting models. Apart from that, however, there seems to be little we agree upon. 
    I disagree that an amplitude-based ('Pythagorean sum') description of the model discussed here are straightforward and easy to explain. To explain Albert's chest using 'Pythagorean probabilities' requires a Mermin-Peres magic square (a 3x3 table containing tensor products of Pauli spin operators). All very elegant, but no matter how you chose to present that stuff, it will not be easy for an average reader to work out that such a contraption of operators indeed describes Albert's chest. In contrast, any 11 year old can sum probabilities (negative or not).


    Secondly, my short answer towards your proposal for "advancing MWI models" is: "shut up and calculate!". 

    My somewhat longer (and more satisfactory?) answer is that my objective here is far more modest than "advancing EPR/MWI models". It is nothing more than an attempt to help the reader building some intuition and appreciation for quantum physics (in relation to EPR/Bell and Kochen-Specker).


    Finally, you are mistaken in thinking that I support "genuine stochasticity". In fact, I expect that some future generation of physicists will classify the stochastics introduced by 'the collapse of the wave function' as the ether of the 20th (21st?) century. But that does not take anything away from the fact that this stochastic description is our (unreasonably effective!) state-of-the-art. Although we all like to digress once-in-while into the realm of our own pet ideas and theories, I consider 'rendering accessible the state of the art in the field' the key task of any science blogger.




    vongehr
    I disagree that an amplitude-based ('Pythagorean sum') description of the model discussed here are straightforward and easy to explain.
    Depends on how you do it. I heard the same about EPR. I tried it anyway, and I succeeded. Don't give up just because others tell you it is impossible.
    Secondly, my short answer towards your proposal for "advancing MWI models" is: "shut up and calculate!". ... attempt to help the reader building some intuition and appreciation
    Your way of building intuition is "shut up and calculate"? Well, since I do not only know that this is completely upside down, but also that you actually do not really mean this (why else would you come up with all the intuitive toy models), ...

    mistaken in thinking that I support "genuine stochasticity". In fact, I expect that some future generation of physicists will classify the stochastics introduced by 'the collapse of the wave function' as the ether of the 20th (21st?) century. But that does not take anything away from the fact that this stochastic description is our (unreasonably effective!) state-of-the-art.
    Johannes - what about getting up to snuff? You are waiting for something that already happened. Why do you think that my models already have randomness fully parametrized? You think it is my private revolutionary idea? Thank you, but no, I also stand on the shoulders of a tower of other small humans. What you write here (God plays dice, collapse of WF, etc) is precisely support of genuine stochasticity!
    'rendering accessible the state of the art in the field' the key task of any science blogger.
    Sorry, but if you want that, you need to first learn the state of the art, and that state is certainly no longer god throwing dice(!!!) but properly parametrized models. Are you actually aware of that relational QM has already resolved EPR? My model is not, as you may presume, a pet idea, but plainly a model that makes their resolution more intuitive! This is the state of the art Johannes.
    Johannes Koelman
    Sorry to say Sascha, but you are throwing out just a a bunch of trivialities. 
    "You are waiting for something that already happened"
    Don't get carried away by your own philosophies and interpretations like MWI. When I say "shut up and calculate" what I mean is that no single "quantum interpretation" invented in the last 80 years has resulted in any observable consequence nor in any hint towards a 'deeper theory'. None of these philosofies and fantasies passes Occam's razor, and like it or not (I personally don't like it), but a conservative strictly operational approach to QM is our current state-of-the-art.
    "Depends on how you do it. I heard the same about EPR. I tried it anyway, and I succeeded. Don't give up just because others tell you it is impossible."
    No others tell me such a thing. This is common knowledge for anyone in the field. Suggest you read up on Mermin-Perez and then reconsider your comment.
    "Are you actually aware of that relational QM has already resolved EPR?"
    There is nothing "to resolve" in EPR. 
    "My model is [..] the state of the art Johannes."
    Let me know when you receive your Nobel.



    vongehr
    Sorry johannes, what about first reading up on what a parameterized theory is at all?
    And there is nothing to resolve in EPR? Really? Wow - you must be more intelligent than all the scientists and philosophers combined. Wonder what they are so worried about. Why not just read your blog - lol.
    "I also stand on the shoulders of a tower of other small humans."

    Could you supply the addresses of a few of the floors of this "tower of other small humans"? Or even some names would be enough to help me see where you stand up there.

    Starting from Sascha, it is crackpots all the way down.

    Names, please.

    I quite enjoyed this entry. The use of negative probabilities is intuitive and easy-to-handle. (In fact surprisingly so. Makes me wonder whether negative probability mathematics had been explored prior to the advent of quantum theory, and moreover why the odds behave so well under such a strange imposition.)

    I actually took to this iteration of your "Quantum Chest" series of thought experiments easier than the prior two. One issue with this specific setup though is that it doesn't give any notion of why you can't open all three rows or columns at once, whereas in real-life the restriction is of fundamental significance.

    Johannes Koelman
    Thanks!

    "One issue with this specific setup though is that it doesn't give any notion of why you can't open all three rows or columns at once, whereas in real-life the restriction is of fundamental significance."

    Right. It would be nice if In this story Albert could in principle open all drawers, but only at the expense of spoiling the measurements. For instance: whenever Albert opens a pattern of drawers that introduces negative probabilities, the chest would blow up in Albert's face. But somehow I feel this would be an unsatisfactory representation of the physics. Another idea I toyed with is to have the opening of a certain drawer to cause all drawers in different rows and different columns to shrink into oblivion. Perhaps a better representation of the physics, but somehow this strikes me as unsatisfactory and too artificial as part of the storyline.


    Happy to hear alternative ideas that would improve the analogy.


    vongehr
    it doesn't give any notion of why you can't open all three rows or columns at once
    Yes, that is why a projection model is much better. If you throw a shadow onto the x-direction (ground), the information about the y-direction (height) is lost (and vice versa). You cannot throw shadows onto both directions at the same time either with only one sun. This is pretty much precisely what happens in QM, namely projection of states onto measurement directions.
    Sadly (for this particular demonstration of the intuitiveness of certain simple macro-scale models of quantum behavior) the "rope and a fork" wikipedia reference has been updated since you first linked to it in that blog post of yours Sascha. It now clarifies via a reference to a journal article describing an actual physical experiment that "Note that the polarization direction is perpendicular to the wires; the notion that waves "slip through" the gaps between the wires is incorrect."

    Nonetheless the mental picture is still helpful, as long as one takes care not to conclude that simple mental analogies imply the hasty conclusion that the analogous quantum behavior is equally simple, or that our macro world intuitions are always trustworthy and can reveal some sort of one-to-one correspondence with the quantum world if only we view them in a certain way.

    vongehr
    You are right - they actually messed the whole thing up - where the hell are the fork and rope gone? Link-rot - I should start uploading everything.
    The problem I have with classical analogies like "shadows casted" is that these don't introduce any real quantum effects (such as violations of Bell's inequalities). Apparently, non-classical behaviors can be enforced in classical models by allowing negative probabilities. Had heard about this before, but never in any detail. Didn't know the idea came from Dirac. All of this begs the question: is a hidden variables description of quantum mechanics possible if one allows for negative probabilities?

    vongehr
    Forget negative probabilities - it is nonsense. The Bell violation does never come from projections, as such can always be classical. It comes from a sort of "extra branching" when different worlds "match up" - however, that is cutting edge so you won't find it here where Johannes still thinks gods throwing dice is a fundamental explanation.
    ~~ Bell violation does never come from projections, as such can always be classical.~~
    My point exactly. I was responding to your reaction above where you state that shadow casting models are better than negative probability models. However, negative probability are capable of describing quantum effects (Bell violations), and shadow casting analogies aren't.

    ~~ a sort of "extra branching" when different worlds "match up." ~~
    Can you make this more precise? As stated, it is way too vague for me. I would be interested to see you derive Bell violations from a branching universe model as simple as Dirac's negative probability model. Would be very convincing to me. In the meantime, hope you don't mind if I stick to Einstein's God casting a die, but ready to jump over to your God branching off new universes. Lol.

    vongehr
    I would be interested to see you derive Bell violations from a branching universe model as simple as Dirac's negative probability model.
    The negative probabilities cannot derive Bell violations; such would be revolutionary. They are a way to do some math more conveniently, but at a high cost: All physical intuition is lost and probability is no longer even anything to do with expectation.
    That the branching allows for Bell violation at the very point where classicality is destroyed can be seen in a completely physical toy model.
    hope you don't mind if I stick to Einstein's God casting a die, but ready to jump over to your God branching off new universes.
    That is precisely the difference! There is no longer any role for anything external (time, randomness, god) in MW models. That is the very reason people already unconsciously reject them. This is not about physics (since MW is completely consistent with QM), but about the human mind being evolved as a reproduction machine in a social environment (agency, ascribing of intentionality, ...).
    ~~ The negative probabilities cannot derive Bell violations; such would be revolutionary. ~~
    Sometimes revolution stares us in the face without us noticing it. Sascha, you have missed the point. Johannes did just that: deriving Bell violations using nothing more than negative probability distributions. He is using the Mermin-Peres variant of the Bell inequalities: http://users.wpi.edu/~paravind/Publications/MSQUARE5.pdf

    Johannes Koelman
    Correct, this is Mermin-Peres (in Kochen-Specker set-up to be precise). Note the above is not new and not my invention. And it is certainly not "revolutionary", lol. There is a ton of publications on Bell violations from negative probabilities (e.g. http://arxiv.org/pdf/quant-ph/0010091.pdf ).
    vongehr
    Sorry Johannes, but you are doing a big mistake here! For example the paper that you suggest, it derives that every LHV model leads to negative probabilities if Bell is violated, which basically is just another way of saying that LHV are impossible! It does also give the negative values that belong to certain Bell violations (OF COURSE, if I make some subtracted m negative, since it is subtracted, all of a sudden the sum is bigger and violates an inequality - that is trivial!). That the paper writes almost as if this means that QM behavior can be derived from "negative probabilities" is crackpottery. Johannes: The archive has lots of crackpottery. Please be more careful. Look at how and what the paper quotes, too! It misrepresents what the big names like Feynman supposedly said! Look also at the paper your other commenter suggested. Look at what that paper writes are "elements of reality", and then compare to what you claim to be similar. The paper the commenter suggests is fine; the paper you suggest here is beyond the edge "fringe science".
    My models are totally consistent with the Mermin stuff, but your negative probabilities are crackpottery straight from the local hidden variables camp. Not quite as bad as Joy Christian's stuff, as far as I can see, there seems to be no conscious cheating here, however, there is certainly a large dosis of misrepresenting at least.
    vongehr
    "X cannot derive Y" is not the same as "Y cannot be derived from X"! I never said that negative parameters as input are not possible, in fact, I already said that they are useful at times. However, there is nothing physical (no phenomenon) from which these negative parameters thus derive the Bell violation!
    BTW: Note that the paper you suggest nowhere talks about negative probabilities and counts all the possible outcomes ("worlds") as "elements of reality".
    Sascha: "X cannot derive Y" is not the same as "Y cannot be derived from X"!

    In other words: "Sunshine cannot result in a wet the pavement" is not the same as "A wet pavement cannot result from sunshine" ??

    It seems you have completely shut yourself off from anything reasonable people are saying here. Bell inequalities are derived (from local realism). Violations from Bell inequalities result (the use of the word 'derived' indicates you don't understand the context) from QM as well as from negative probability theories. These latter theories violate local realism as only non-negative probabilities (for instance the averaged variables yielding the semi-classical limit) can correspond to elements of reality.

    I am probably wasting me time and energy here. Will no longer follow this thread.

    vongehr
    Sorry - you need to read more carefully what people actually write and not "read" what you think to know they write because you have them already in a drawer.
    You can input something like the totality of many worlds, thus come to certain mathematical expressions, and those then help to derive violations of inequalities.
    But you cannot just have violation, then find that the mathematical tools, like probabilities, become nonsensical (here negative), and instead of concluding that one of your assumptions (e.g. hidden variables) is wrong, simply go ahead and present the issue as if the violation is derived from starting out with nonsensical math. You can derive anything from starting with nonsense math. That is not how proper physics is done, that is how you do crackpottery.
    Ha, ha, vongehr is revealing Dirac and Feynman to be crackpots. Hilarious.

    On a more serious note: this new article on the physics of negative probabilities ( http://arxiv.org/abs/1210.6870 ), by Imperial College and Cambridge physicists, Halliwel and Yearsley, I found quite intriguing.

    lumidek
    A beautifully written post. Still, the actual state of the chest could be useful to write down explicitly. ;-)
    Wouldn't you agree to repost it as a guest blog on my website?

    http://motls.blogspot.cz/?m=1
    Johannes Koelman
    Hi Lubos. Thanks for the kind words. No problem if you repost, but appreciate if you incorporate a link to this page. (I desperately need more non-crackpot traffic... Can your site bring me that? ;) )
    lumidek
    Thanks a lot. Please check here:
    http://motls.blogspot.cz/2012/11/quantum-casino-less-than-zero-chance.html
    http://motls.blogspot.cz/2012/11/quantum-casino-less-than-zero-chance.html?m=1

    I know that very many non-crackpot readers visit my website but most of them stay silent. 
    BTW do you have an explicit formula for the state vector of the 9 qubits that leads to the observations?

    Update: I probably know how to do that. They're not 9 independent qubits, of course, because then they would behave classically. But one may create the 9 operators as products of some Pauli matrices/spins so that the even/odd number of the factors anticommute with those of their friends in the rows/columns.
    Johannes Koelman

    "But one may create the 9 operators as products of some Pauli matrices/spins so that the even/odd number of the factors anticommute with those of their friends in the rows/columns."
    The 3x3 table of operators you are looking for consists of direct products of Pauli matrices with operators in rows mutually commuting, and similarly for those in columns. It's called the Mermin-Peres magic square. 
    blue-green

    “Shut up and calculate” works fine until one gets tripped up calculating. I followed your link Motl and noticed your recent excellent post on Feynman sums and diagrams. The mathematics and physics quickly gets very hairy. A graduate level lecture on Feynman sums from a department of mathematics would be unrecognizable from one in a physics department where prickly issues on divergence and convergence and what to count (and the underlying spaces and fibres) are glossed over.

    Maybe we have this whole outreach program upside-down. Maybe it's the classical world that's the weirdo that's hard to fathom ~ not the quantum world.

    The quantum fundamentals are not so difficult and students learn them with basic atomic chemistry. There are only a few dozen frequently encountered elements from the periodic table. There are even fewer common elements at the level of quarks. Each of these types of elements has no individuality. Unlike students in a classroom, they can swap places with no observable effects. Their characteristics are also completely independent of the place and time in which they are observed.

    Contrast this quantum simplicity with the classical mess in which everywhere and every when there are petty individual differences burdened with history, cliques and unfairness in the way things are counted and tallied. It's the classical world that is the hydra-headed monster, not the quantum world.

    The problem with the stick-and-ball models of molecules is that students tend to think of the balls as being like little classical balls with each one being a distinct “particle”. That may be OK for understanding how pressure works in a balloon, yet it is useless for understanding solid state physics or phase changes .... or entanglement.

    Instead of trying to make quantum physics seem classical … and instead of pretending that its calculations are easy, how about a different approach that doesn't slip in assumptions about an underlying space and time that isn't there?

    Johannes Koelman

    "Contrast this quantum simplicity with the classical mess"
    It is relevant to note that this "classical mess" is the emergent behavior we are presented with following some significant coarse-graining. Unfortunately, this emergent behavior heavily colors our idea of reality. Deep down reality is much more austere than the "classical mess" suggests.
    blue-green

    Although “course-graining” has its uses, it may well be that 140 years after its use in Boltzmann's H-Theorem, it has been mined for all it is worth.

    And yes, if one squints hard enough, quantum mechanics may appear to be “austere”. 

    However, when one considers many-body systems, the dynamical space for a many-body system is an extremely high dimensional cross-product of all of the individual spaces available to each “body” …. whereas …. in the blurred over and “austere” classical budget-vision, one simply adds more particles to a single box … with no window into biology or consciousness, even if you do get an emergent space and time.