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Entanglement: why two schoolkids always answer questions oppositely

More than a week ago, I discussed an article by Natalie Wolchover who was apparently shocked that when some optical data from the stars are used to produce pseudorandom numbers, an experiment testing entanglement with some random choices for the detectors produces the same results as the experiment where only terrestrial gadgets are used as the pseudorandom generators.

What a surprise: numbers that look like some random mess with the same distribution lead to the statistically identical outcomes whether or not they were calculated from stars or dice. Come on, people. This is totally basic common sense. There can't be any correlations of the terrestrial experiments with the random stellar data. To believe that there are such correlations – that the experiment cares whether the stellar data were employed – isn't just analogously silly as astrology. It really is a special example of astrology! This is what astrology really means: local events on Earth do care about some immediate properties of the celestial bodies! Well, they don't. None of the data from local, repeatable experiments on Earth can be correlated with some independent data about the celestial bodies.

You may also say that the belief in these correlations with the stars is on par with the Movie Pi where the digits of \(\pi\) were assumed to know all the information about the movements of the stock markets and prophesies of the Jewish Bible, among other things. Please, give me a break. It may be an inspiring movie but everyone who has spent at least some time by looking at the actual relationships between events in the world, not necessarily the "physical laws" in the narrow and technical sense, must know that this is the kind of a relationship that cannot exist and elementary evidence is enough to justify this assertion.

Now, an appendix to Wolchover's article about the stellar entanglement conspiracies (that were "surprisingly" not detected by an experiment)

How to Tame Quantum Weirdness
was written by Pradeep Mutalik, a writer who was previously mentioned because of a confusingly ambiguous article about the Sleeping Beauty Problem. The title talks about taming of quantum weirdness but I think the actual purpose is to spread the illusion or delusion that quantum mechanics and the entanglement are weird.




What's going on? In the simplest entanglement experiment with two spin-1/2 particles, one starts with the \(J=0\) (singlet) state of the two spins\[

\ket\psi = \frac{ \ket\uparrow \ket \downarrow - \ket \downarrow \ket\uparrow }{\sqrt 2}.

\] When the measurements of the projection of the spin along the same axis \(\hat n\) is performed on both particles \(A,B\), we find out that the eigenvalues which may be either \(+1/2\) or \(-1/2\) are always exactly opposite to each other:\[

J_{A,\hat n} = -J_{B,\hat n},\quad \{J_{A,\hat n},J_{B,\hat n}\}=\{+1/2,-1/2\}.

\] This perfect anticorrelation simply follows from the conservation of the angular momentum. The initial state is an eigenstate of\[

\vec J = \vec J_A + \vec J_B

\] with the eigenvalue zero (vector) – it is possible for the vector to be an eigenstate of all three components in this special case when the eigenvalues are zero. So when you measure a component of \(\vec J\), namely \(\vec J_{\hat n}\), you are guaranteed that the two terms exactly cancel each other.




This outcome holds for any choice of the axis \(\hat n\). The fact that the initial state preserves the perfect anticorrelation regardless of the choice of \(\hat n\) – regardless of the rotation we perform on the system – mathematically depends on the fact that the singlet state above has the structure of the antisymmetric spintensor \(\epsilon_{\alpha\beta}\) where \(\alpha,\beta\in\{1,2\}\) or \(\{\uparrow,\downarrow\}\), if you wish, are spinor indices which is an invariant under the \(SU(2)=Spin(3)\) rotations. Under a rotation, each spinor index \(\alpha,\beta\) is rotated via an \(SU(2)\) transformation but the epsilon symbol remains constant because it transforms as a determinant of the \(SU(2)\) matrix and this determinant is one, you know, because of the letter \(S\) (for "special") in \(SU\) ("special unitary").

What is Mutalik's "model" for that?
Two students, \(A\) and \(B\), answer one test question, a Yes/No question, in a long exam. And they just happen to answer oppositely every time, whatever they are asked about.
All the details that Mutalik adds – that some exams are done after 37 days, or questions are grouped into groups of 100, and so on – seem like absolutely irrelevant distractions. Now, he asks you to decide whether it's reasonable that all the answers will always be anticorrelated.

Well, if they were answering randomly and independently, the anticorrelation would almost certainly disappear very soon. But that doesn't mean anything mysterious or weird. It just means that the model in which two students answer randomly and independently is completely different than the situation of the two spins in the singlet state.

The actual reason why the answers are so perfectly anticorrelated in the experiment with the two spins is that the two spins were prepared in the singlet state that encodes the perfect anticorrelation – e.g. because the electron and the positron arose from a decay of a spin-zero particle (two photons coming from a decaying positronium would be easier for that). Mutalik doesn't discuss the preparation of the entangled state at all which means that he fails to see absolutely everything about the actual reasons behind the anticorrelation.

A closer model for the anticorrelation would be like this.
Every morning, before they go to answer a question in the exam, Alice and Bob agree to give the opposite answers to a question. However, the answers must look random and they must be dependent on the question that is being asked. How do they do it? Alice and Bob agrees that if the third letter in the question they're asked on the exam later today is a vowel, Alice answers Yes and Bob answers No, and if it is not a vowel, Alice answers No and Bob answers Yes.

They go, answer their question, and what a shock: The answers are anticorrelated. Every day. For example, they are asked: "Will Donald Trump make America great again?". Alice answers No and Bob answers Yes because "L" is not a vowel.

On the following day, they may choose the seventh letter of the question as the source of the information. Or some other binary information extracted as a function of the question.
There is nothing mysterious about the anticorrelation: the anticorrelation was guaranteed from the beginning when \(A,B\) were in contact and the probability that the answers are anticorrelated was 100% from the very beginning. In the actual quantum mechanical situation, the answers of Alice and Bob are perfectly opposite but they hide one bit of information, e.g. the answer by Alice, that is truly and perfectly random. This isn't really captured too well by the model which emulates the random decision by some shared plan of Alice and Bob "how to answer".

But otherwise my model has everything it needs. The answers are perfectly anticorrelated, just like in the case of the two spins. And they are anticorrelated for the same reason – the anticorrelation was pre-programmed into both subsystems while they were in contact. Also, my model agrees with the quantum mechanical reality when it comes to the "pre-existence" of the individual answers. After Alice and Bob say good-bye to each other and are moving apart, it is not determined yet whether Alice would answer Yes or No (i.e. whether Bob would answer No or Yes) because the actual answer will depend on the question that is being asked and the question may be changed right before it is asked!

Most importantly, there is no superluminal communication and no action at a distance in either case. All the correlations or anticorrelations are explained by the two subsystems' contact or "agreement" at some point in the past when their separation was zero or tiny.

Now, my model would work perfectly for all tests in which Alice and Bob are asked exactly the same question: the only facts you need to reproduce is that Alice's answer seems random – Yes and No are equally likely and there are no patterns if you repeat the experiment many times – and Alice's and Bob's answers are perfectly anticorrelated. But if I allowed experiments in which Alice and Bob are asked different questions, my model won't be able to produce all the correlations implied by quantum mechanics.

When the axes \(\hat n_A\) and \(\hat n_B\) for the two spins' measurements are chosen "almost the same" but not quite, it's still vastly more likely that the two outcomes will be anticorrelated. But in my model, if Alice and Bob are asked similar but slightly different questions, it's much less guaranteed that the anticorrelation will be so good. Alice may be asked: "Will Donald Trump make America great again?" And Bob may be asked: "Trump will lower corporate taxes to less than 20%: Yes or No?" These two questions are almost the same and they should still be giving "mostly anticorrelated" answers. But their algorithm based on the vowel in the third letter will make them give the same answer in this test instead.

So in details, when all possible experiments with any choice of axes are picked, my model won't work like quantum mechanics. Indeed, Bell's theorem guarantees that no local realist model may give the same answers as quantum mechanics. The rules of quantum mechanics "know" how the measurements with respect to nearby axes should be "similar" while my vowel-based model or any local realist model will fail to correctly account for the proximity of nearby questions in generic cases. The precise way how the probabilities of answers to different questions are determined according to quantum mechanics is simply different than the way adopted by any local realist theory. Quantum mechanics isn't a realist theory. In the same way, radio is something else than a war pigeon and the details how they transfer the information differ from one another, too. The difference is not shocking, it is not a big deal.

But the broader logic how the information is propagated and "when" the correlations or anticorrelations (if any) are determined, those qualitative things are perfectly captured by my local realist model (my model is a hidden-variable model, the hidden variables remember the algorithm how to deal with the vowels etc.). The anticorrelation of Alice's and Bob's answers results from their agreement that guarantees that they are going to give the opposite answers. There's no action at a distance: it's simply a result of a discussion between them when they could hear each other. And if no corresponding (direct or indirect) contact of the two subsystems existed in the past, there can be no correlation or anticorrelation!

I really don't understand what so many people find so difficult about these straightforward things. And I don't understand why the Quanta Magazine encourages Mutalik – who is clearly one of those who don't understand quantum mechanics – to pick the winner who has given the "best model". By the way, Mutalik's model involves some inflated balloons with anticorrelated but random red and blue sub-balloons. Some parts of his "model" are equivalent to my model. But why is he joining the two balloons by a rope? There is no rope between the two subsystems. They don't communicate when they're measured. They just pre-agreed to give answers according to an algorithm or rule that guarantees the anticorrelation when the same question is asked to both!

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