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.