Thursday, June 25, 2015

The socks of Mr. Bertlmann


It seems that I created quite a stir with my prior post and despite knee jerk emotional rants to the contrary which were mostly absurd misunderstandings like I am secretly a believer in classical physics, what I said there is still completely correct (up to grammatical mistakes and typos). One point of genuine disagreement however were about the well known paper of Bell: "Bertlmann's socks and the nature of reality" which I discovered it is greatly misinterpreted and misunderstood. There were other genuine disagreements which I will get to in future posts but I can only address one issue at a time. Today I will try to explain the Bertlmann's socks paper in the larger context of Bell's results.

Let me first set the stage. From its discovery, quantum mechanics was a constant source of debates and disagreements. Einstein had a great dispute with Bohr, Schrodinger did not like quantum mechanics implications and he concocted his famous cat in the box example. Less known is the position of Karl Popper, the discoverer of the falsifiability criterion. In 1959 Popper was trashing Heisenberg's uncertainly relations. His point was that the uncertainty relations correspond to physical characteristics after measurement and in principle there is no precision limitation to defining the position and momenta of a particle and so in his opinion Mr. Heisenberg was unnecessarily jumping to conclusions in his positivist approach. Then he said the following (this is a translation from English to Romanian and back to English so the original quote may be sightly different, but the meaning is clear enough):

"Because any proof of this kind must use quantum theory considerations applied to individual particles, hence formal probability statements, this must be translated word for word in statistical language. If we do that, we'll see that there is no contradiction between the particular measurements assumed to be precise and quantum theory in its statistical interpretation."

Why is this important? After all Popper is not know today to be a quantum guy. However back in 59 he was quite influential developing his own interpretation of quantum mechanics and the fact that he is not known today is because he was wrong and naturally got forgotten. But people today sometimes state that Bell's inequalities were already old news and Bell did not do much. My point is simply that around that time people were not aware of of those inequalities and Bell's results came as a shock.

So Bell put Popper's nonsense to rest with his result and showed that there is a contradiction in statistical terms between any local realistic theory and quantum mechanics. How? By the use of his correlation inequality. Bell had several motivations and today I will present his ideas from one particular point of view skipping the usual discussion with von Neumann. Bell started the analysis with the Bohm and Aharonov variant of the EPR gedankenexperiment in which a source of electrons emits pairs of electrons in a total spin zero state:

\( |\Psi\rangle = \frac{1}{\sqrt{2}}( |up \rangle_{left} |down \rangle_{right} - |down \rangle_{left} |up \rangle_{right} )\)

Measuring the spins for the left particle on direction a and for the right particle on direction b yields the correlation \(-a \cdot b \) or minus the cosine of the angle between the two measuring directions. Can this be explained if the spins had pre-existing values before measurement? If the measurement directions are perfectly aligned, anti-aligned, or orthogonal, from total spin conservation it is easy to predict that the measurement correlations would be -1, 1, and 0 no matter what. And what would happen if the two electrons would have the spins on opposite directions to preserve the total spin zero state, but their spins would be randomly distributed in space? After about a page of an integration exercise you can convince yourself that the correlation would be in this case \(-\frac{1}{3}a \cdot b \), so case close, right? Bell arrived at this -1/3 result too but he did not like it enough to ask to be put to an experimental test and he looked further. He noticed that the slope of the correlation curve is zero when the directions are parallel and that looked strange.

Can he arrive at this kind of correlation curve \(P(a, b)\) while assuming that the outcomes A for Alice and B for Bob depend only on the local measurement direction (no superluminal signaling), on some hidden variable \(\lambda \) and (very important) respecting the factorization condition below?

\(P(a,b) = <A(a, \lambda) B(b, \lambda) >\)

where the angle brackets mean average over \(\lambda \). This factorization is the famous Bell locality condition in which the outcomes depend only on the local physics (the directions a and b in the local laboratories) and on a shared randomness "hidden variable" \(\lambda \) assumed to be generated at the moment of the emission of the two electrons.

So Mr. Bell discovered that for any theory obeying the factorization condition from above he would not get a zero slope correlation curve but a "kink". See the picture below from another Bell paper entitled: "Einstein-Podolsky-Rosen experiments"


Also from the factorization (Bell's locality) condition from above it is not hard to obtain Bell's original inequality:

\(1+ P(b, c) \geq |P(a, b) - P(a,c)|\)

But what does this mean and why is the correlation slope flat for quantum mechanics and is a straight line for classical physics (which does obey Bell's locality condition). The key is in the factorization or lack of. Take a look at the singlet state wavefunction from above. You cannot factorize it between the left and right particles and you do not get the straight line correlation curve. The existence of the flat curve of quantum mechanics requires a different explanation. Enter the Bertlmann's socks paper now.

There are several Bell inequalities, and quantum mechanics and Nature does violate them. But why? The key pedagogical simplification came from Bernard d'Espagnat which came with this silly but true statement:

"The number of young women is less then or equal to the number of women smokers plus the number of young non-smokers"

Let's explain this better with Venn diagrams:


and let us call Women the set A, Non-smokers the set B, and Old the set C. Then the statement reads:

A and not C <= A and not B + B and not C 

Is this true? Let's check:

A and not C = areas 1+6
A and not B = areas 1+2
B and not C = areas 5+6

A and not B B and not C = areas 1,6,2, 5 which is larger or equal with the areas 1 + 6 (equal when the areas 2 and 5 contain no elements).

So far so good, but what does this have to do with quantum mechanics and Nature? Mr.Berltmann enters now the stage:


Dr. Bertlmann was an eccentric person who was always wearing socks of different colors. As soon as you see the color of one of his sock you know the other one is not the same. Now in this case the socks have definite colors before you look at them which is different than the spin direction in the electron case which does not exist before measurement and this is the key difference. Can we put this in an exact mathematical statement and more important, can we test this in an actual experiment to show electrons are not like the socks of Dr. Bertlmann?

Now back to d'Espagnat, thank you Dr. Bertlmann for providing humor to a serious physics, mathematical, and philosophical problem.

When a characteristic (be it color of socks, gender, smoker status, color of eyes, etc) exists independent of measurement then the natural way to describe it is using the concept of a set because you can perform the simple test of belonging to your set or not and the result in unambiguous: you are either inside the set, or you are outside. You are either a smoker or you are not, you are male or a female, etc.

Sticking with socks for now, Mr. Bell considered 3 sets, A, B, and C as follows:

A=the number of socks which survive 1000 washes at 0 degrees Celsius
B=the number of socks which survive 1000 washes at 45 degrees Celsius
C=the number of socks which survive 1000 washes at 90 degrees Celsius

Then following the Venn diagram from above he considered if :

A and not B + B and not C >= A and not C 

which would be true. But does this inequality hold for electrons as well? You cannot "wash 1000 times an electron at 45 degrees Celsius", but you can detect if the spin records up when measuring it with a Stern-Gerlach device oriented at 45 degree angle. So if the spin orientation of the electron exists independent of measurement we can have the following 3 sets:

A=the electron records spin up when passing through a Stern-Gerlach device oriented at 0 degrees
B=the electron records spin up when passing through a Stern-Gerlach device oriented at 45 degrees
C=the electron records spin up when passing through a Stern-Gerlach device oriented at 90 degrees

Sure, but what to do about this business of "A and not B". You cannot pass at the same time through two detectors! But here is the trick: you have two electrons in the singlet state. Moreover you know that no matter what direction you chose for the left detector, if the right detector is opposite aligned, both detectors will record the same answer because of the total spin conservation. Therefore "A and not B" means now that the left particle clicks up when measured at 0 degrees, and the right particle clicks up (which from spin conservation is equivalent with the left particle clicking down or the left particle not clicking up) when measured at 45 degrees. Sure, there is a bit of counterfactual reasoning, but it works.

So now we have another genuine Bell inequality:

the number of [left electrons clicking up when measured on 0 degrees and right electrons clicking up when measured on 45 degrees]
+
the number of [left electrons clicking up when measured on 45 degrees and right electrons clicking up when measured on 90 degrees]
>=
the number of [left electrons clicking up when measured on 0 degrees and right electrons clicking up when measured on 90 degrees]

And those 3 numbers can be easily computed using quantum mechanics and the answer is...

\(\frac{1}{2} \sin^2(22.5) + \frac{1}{2} \sin^2 (22.5)  \geq \frac{1}{2} \sin^2(45)  \)

or

0.1464 >= 0.2500 !!!!!!!!!!

And guess what? Not only quantum mechanics violates this inequality, Nature does it too just as quantum mechanics predicts it does.

So what happened? How can this be true? In quantum mechanics sets and Boolean logic do not apply. When you measure something in quantum mechanics you project to a subspace of the Hilbert space and the Boolean logic changes to the logic of projections. When a system has a property like say spin this is not representable as a point in a set. The Venn diagrams have to be generalized from flat circles in a plane to subspaces and their intersection is not as naive as in the picture above. Quantum OR and Quantum NOT are very different than classical OR and classical NOT. All this is because of the novel property of superposition which does not exist in classical physics. Superposition is what makes the Hilbert space a relevant mathematical description to what is going on.

And this is the business of Bertlmann's socks paper.

Now back to the misuse and misunderstandings of this paper. Last time I stated:

"[I cannot take Schack's Bertlmann comment at face value as this would imply he disagrees with Bell's mathematical statements from his famous Bertlmann's socks paper and that would be wrong]."

to which Lubos Motl objected. When you state that "quantum correlations are like Bertlmann socks" at face value you state that there are no differences between classical and quantum correlations and that the difference between the kink vs flat curve of correlations is not there. The big point of Bertlmann's socks paper is that quantum and classical correlations are fundamentally different. And this is not me stating it, it does not come from a faulty understanding of the paper, but it is stated by Bell himself in the very first sentences of the paper and you cannot get more explicit than that:

"The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein-Podolsky-Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann's socks is often cited."

If the correlation curves are not fundamentally different, then you can create classical models of quantum effects, which in turn means that the spin has a definite orientation before measurement. But I know Schack does not believe that because he always emphasizes the importance of Kochen-Specker theorem. The right way to understand his statement was as I stated before:  quantum correlations are just correlations and no explanations are needed in general and I agree with this point of view because there is no way to explain them by reduction to hidden variables which is the content of Bell's theorem. [My position is a bit stronger than what QBism advocates. QBism appeals to the trip between Alice and Bob needed to be able to compute the correlations and this makes perfect sense in their approach. I however say respect nature for what it is and just stop whining about the lack of an explanation to appease your classical intuition which is the result of biological evolutionary pressures.]

But stating it like this: "quantum correlations are like Bertlmann's socks" invites protests and follow up clarification questions from the people who do understand very well the Bertlmann's socks paper. In other words, it adds spice to conversation and it is a provocation for reaction, a friendly poke aimed at the Bell experts who may also (but not always-I am a counterexample and I am not alone) believe in something more: beables. But beables, the unfinished research project of Bell, are a topic for another time.

Also, back to Bell's factorization condition. This is called Bell locality and next time I'll dig into it some more. Nature violates Bell locality precisely because nature is quantum mechanical and not classical mechanical. It does not mean you can send signals faster than the speed of light and violate relativity. If you have a problem with the name you are not alone, but you are in a minority, tough luck, this is a standard term now. If you want to change it, do something really important in the foundations of quantum mechanics on par with what Bell did and then rename it to whatever you like. Calling the foundations community idiots leads nowhere.

Side announcement: I will be going on vacation for a week tomorrow and I will not have internet access. Therefore I will not be able to read or reply to reactions about this post. My next post will also be a bit delayed.

Update: I just came back from a trip to Alaska and I'll need a couple of days to get up to speed and write the next post. You can expect it at the end of Monday.

12 comments:

  1. Quantum correlations are just special examples of correlations - and correlations are a general concept not linked to any physical theories. They were used long before quantum mechanics was born and quantum mechanics hasn't changed what the term "correlation" means. Quantum entanglement is the most general pure state description of the state of composite systems that predicts correlations.

    You clearly misunderstand 100% of this stuff when you don't get this point. Entangled electrons have exactly the same reason to be correlated as Bertlmann's socks: mutual contact in the past. Quantum entanglement have a more general, advanced formalism to be described than correlations that are predictable from classical physics. But at the end, in the real world, even Bertlmann's socks are an example of *quantum entanglement*. There is no difference because the correct description of *all* correlations in the world around us is the quantum description, i.e. in terms of entanglement. That's true as long as we know the pure state. If we describe the system by a density matrix, the correlations in QM become even more indistinguishable from the correlations in classical physics. But they always demand a quantum description if one wants to be accurate. Classical physics is always wrong and it's wrong for a precise description of socks, too.

    Even more obviously, it is complete nonsense - as you state - that quantum mechanics violates the laws of Boolean logic. Quantum mechanics tells us that, just like classical physics, one can make Yes/No statements about Nature - about observables. These statements or propositions are propositions in the usual sense of logic and obey all the rules of the logic. Quantum mechanics just gives us a different recipe than classical physics to - unavoidably probabilistically - predict the value of new propositions from the knowledge of the old ones. In some formalisms like the consistent histories, the condition that all propositions will violate the rules of logic - the usual laws for addition of probabilities etc. - is a consistency condition that is imposed e.g. on the histories. But when it is imposed, and it has to be, the rules of logic always hold.

    "Calling the foundations community idiots leads nowhere."

    On the contrary, this is the first, totally vital step that leads somewhere - to the elimination of this crackpot movement from science where they simply don't belong, and restoring the scientific method in these corners of science that should become physics once again.

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    1. "all propositions will violate the rules of logic": "violate" should have been "respect"

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  2. Lubos, I have only 4 hour until my flight and then I'll be out of internet access for a week so I'll give you a very quick answer for this:

    "Even more obviously, it is complete nonsense - as you state - that quantum mechanics violates the laws of Boolean logic"

    You are 100% wrong on this and I will blog about it too in detail to explain just like I did with the socks paper-but this topic will have to wait for its turn.

    In the meantime read this:
    http://www.mathematik.tu-darmstadt.de/~ziegler/qlogic.html and take a look at the image above references.

    this is what I could quickly find on the internet that is semi-understandable, sorry for the bad background. Also read the book of Beltrametti and Cassinelli: http://www.amazon.com/The-Logic-Quantum-Mechanics-Encyclopedia/dp/052116849X.

    The field started with the works of Birkhoff and von Neumann, continued with the key result of Piron (https://en.wikipedia.org/wiki/Constantin_Piron) and is now extended by his student Aerts http://www.vub.ac.be/CLEA/aerts/research.html among others

    There are entire conferences dedicated to the lattice approach to QM, it is a big community out there that also includes logicians and mathematicians.

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  3. Dear Florin,

    You wrote:

    "Nature violates Bell locality precisely because nature is quantum mechanical and not classical mechanical. It does not mean you can send signals faster than the speed of light and violate relativity."

    I think you are too quick to dismiss classical mechanics. Bell's theorem requires some assumptions which are not true for the a certain class of hidden variable theories, classical deterministic field theories.

    1. These theories do not allow counterfactual reasoning (CR) at all.

    2. Even if we allow for CR, detector orientation and the spin of the entangled particles cannot be treated as independent parameters. The trajectory of each particle is a function of all other particles' position and momentum. This is true for the entangled particles but also for the particles in the detectors. A change in the detector orientation means a change of the position of the particles in the detector which produces a change in the local field experienced by the entangled particles. As a result, the force acting on the test particles changes so their trajectory changes as well. This is also an obvious explanation for the "mystery" of the double slit experiment. Two holes correspond to a different particle distribution than one hole, so the test particle feels a different force in these two situations, leading to different trajectories and ultimately to different patterns on the screen.

    Andrei

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    1. Dear Andrei,

      Despite my belief in quantum mechanics I try to maintain an open mind because personal beliefs are irrelevant in physics. If nature is classical at core, so be it, I only care to be correct in describing nature.

      However I am not aware of the results you spoke about. Can you point me to same papers or archive preprints? You can do do it here or privately if you desire: fmoldove@gmail.com

      Best,

      Florin

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    2. Florin,

      In my previous post I've pointed out that there is no good evidence for an incompatibility between the predictions of quantum mechanics and classical deterministic field theories. I have tried to justify why the assumptions used in Bell's theorem or Free-Will theorem fail for this kind of theories.

      As far as I know the only physicist actively developing a classical interpretation of QM is ’t Hooft. I fully recommend his papers:

      The Free-Will Postulate in Quantum Mechanics

      The Cellular Automaton Interpretation of Quantum Mechanics. A View on the Quantum Nature of our Universe, Compulsory or Impossible?

      They can be found here:

      http://arxiv.org/pdf/quant-ph/0701097.pdf
      http://arxiv.org/pdf/1405.1548.pdf

      His work is mostly directed towards discrete models, but his arguments against the the idea that hidden variable theories have been ruled out are very well presented.

      Andrei

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  4. Andrei,

    Touche, you are right! Yes, superdeterminism is a way out and I am familiar with ’t Hooft's approach. Therefore to be completely correct, I need to qualify my post from above by stating that what I said is true up to superdeterminism.

    However, complete lack of free will couples QM with the rest of physics in one huge ToE and you have to explain now much more than QM. This may be a good thing if you are after solving quantum gravity problems, but you need to explain the gauge groups of the Standard Model as well. So you need to do something on par with string theory and that is very tall order.

    I have three biases against superdeterminism. First, I know how to obtain QM from natural postulates and explaining QM from CM violates Occam's razor. Second: at the moment I know no viable alternative to string theory. Third: "I know" I have free will.

    In theory superdeterminism is a possible approach (and as a result I stand corrected), but in practice (at least for me given limited time and resources) I don't think is a viable alternative and that is why I did not mention it earlier.

    Florin

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  5. Florin,
    Now I understand that the dispute about ‘locality’ is just semantics, i.e. unfortunate use of that word by Bell for non-factorizability. I am surprised that, as you say, that is a common usage. Locality means "signals not faster than light" for just about everyone believing in relativity. Relativistic field theory is called “local field theory” the world over. It may be even that perhaps at that time Bell was having doubts about validity of relativity. What do you think?
    As for the other issue on Bertlmann’s socks, I am not sure if that is also semantics problem. BTW, interestingly, Bertlmann’s article just appeared in July issue of Physics Today. Do you agree with all the statements made there?

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  6. Kashyap,

    Thank you for pointing me to the article in Physics Today. I gave it a quick glance but I have to carefully read it to see if I spot anything wrong there.

    Bell had never have any doubts about relativity, it was a later careful analysis of what happened in his theorem which made him arrive at non-locality as the main argument against EPR.

    Bell has this strange paper on how to teach relativity where he looks at relativity in a funny nonstandard, pre-Einstein way to arrive at some intuition of why a rope between two uniformly accelerated spacecrafts must break. But he did that because in an informal poll at CERN he found out that almost everyone got the wrong answer, not because he did not know relativity or he did not believe in it.

    About nonlocality, the term started with Bell's analysis and the term stuck in the foundations without anyone from within challenging it. Early on it was a time of "circling the wagons" and defending against outside criticism and this is how the term got firmly established. I like to cal it something else but I don't have enough clout for that.

    There is also another confusion about locality. In quantum field theory you have microcausality and this is used as an argument for locality in QM. This is incorrect reasoning because microcausality is outside the scope of QM axioms: it belongs in Wightman's axioms of QFT. QFT is not QM; informally speaking it is QM+relativity.

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  7. OK! QM is not QFT! But I doubt if there is a single physicist around who believes in QM but not in QFT! People may believe in generalization of QFT though. You may have to convince others to drop the word "locality" for non factorizability.

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  8. "I doubt if there is a single physicist around who believes in QM but not in QFT!"

    Indeed.

    "You may have to convince others to drop the word "locality" for non factorizability."

    Not me. I want to replace the non-locality term, not the locality one.

    By the way, read my next post which I'll publish tonight. I'll explain this business of non-Boolean logic in QM. Dualities don't only exist in sting theory, there is a duality of QM description in terms of non-Boolean logic:

    the standard Hilbert space formulation of QM --- a particular non-Boolean logic

    Due to this duality the particular non-Boolean logic has a unique geometric representation which is the statement of Piron's famous theorem. Propositions in this non-Boolean logic are k-dimensional subspaces of the Hilbert space. Much cooler ways to reason than using Venn diagrams in classical Boolean logic.

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