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Contextuality of our quantum world is a proven fact

In a recent article reviewing Feynman's Messenger Lecture on quantum mechanics, an anti-quantum zealot reader complained that ordinary hidden-variable theories and Ghirardi-Rimini-Weber "random collapse" theories are an easy target. They're just straw men: what about "contextual hidden-variable theories"? Well, a problem is that there aren't any.

But in this text, I want to settle the claim that all non-contextual hidden-variable theories are dead. What does it mean for a theory to be "contextual"? It means that there can't objectively exist any prepared answer what a looming measurement of any kind will reveal – an answer that would be independent of the type of measurement that the experimenter chooses to do. Those who assume that the world "has to be" non-contextual may be said to be perpetrators of the separation fallacy.

Quantum mechanics is clearly contextual in this sense; hidden-variable theories that have been constructed are non-contextual in this classification. They include an objective reality that is independent of the context. And they're therefore excluded. Feynman's Messenger Lecture on quantum mechanics showed that the good old double-slit experiment is enough to prove that all non-contextual theories inevitably disagree with the basic experiments: if the "which slit" information were ready for the case that we decide to use photons and measure "which slit" the electron took, then it would be guaranteed that there can't be any interference pattern in the case when we allow the electrons to interfere.

However, the hidden-variable apologists and anti-quantum zealots in general are deaf and blinded so the research continued and even these days, people publish new papers that claim that "finally, we can settle this debate" and show that non-contextual theories are wrong.

Here, I want to look at a 2009 paper in Nature,

State-independent experimental test of quantum contextuality (arXiv)
by Kirchmair et al. (A popular intro was written by Boris Blinov, also in Nature.) Using trapped ions, they falsify all non-contextual theories in physics. Their proof doesn't depend on the assumption that the experiments have to be very precise. Indeed, their error margin is nonzero but it's still enough to settle this question.

Let's consider a four-dimensional Hilbert space that may be represented as the tensor product of two two-dimensional Hilbert spaces of a spin-1/2 particle,
\[ {\mathcal H} = {\mathcal H} (\vec s_1) \otimes {\mathcal H} (\vec s_2) \] Linear algebra allows us to play lots of funny games here. But we will define the following \(3\times 3=9\) observables
\[ \left(\begin{array}{ccc}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{33}\\ A_{31}&A_{32}&A_{33}\end{array}\right)= \dots \] as various Pauli matrices or their tensor products:
\[ \dots = \left(\begin{array}{rrr}\sigma_z \otimes 1\,\,\,\,&1\otimes \sigma_z&\sigma_z\otimes \sigma_z\\1\otimes \sigma_x&\sigma_x\otimes 1\,\,\,\,&\sigma_x\otimes \sigma_x\\ \sigma_z\otimes \sigma_x&\sigma_x\otimes \sigma_z&\sigma_y\otimes \sigma_y\end{array}\right) \] The first (left) matrix in the tensor products acts on the first spin's Hilbert space, the second (right) matrix acts on the second one.

Now, let's look at these nine operators. You may easily check that any pair of operators that are written in the same row commute with one another; any pair of operators in the same column commute with one another, too. So you may measure the triplet in a given row simultaneously; and you may measure the triplet in a given column simultaneously, too. Also, the product of three operators in each row is equal to \(+1\); the product of three operators in each column is also \(+1\) with the exception of the third (last) column whose product is equal to \(-1\).

This sign is shocking because the eigenvalues of each of the nine operators are equal to \(\pm 1\). Any non-contextual theory must prepare nine numbers \(A_{ij}\) prior to the measurement. Each of them is equal to \(\pm 1\) because no other result may be obtained. Because the products of three quantities in each row and column, denoted
\[ (R_1, \, R_2,\, R_3,\,C_1,\,C_2,\,C_3) \] are equal to \((+1,+1,+1,+1,+1,-1)\) in quantum mechanics, according to the definition of the operators \(A_{ij}\), we also have
\[ \langle R_1\rangle + \langle R_2\rangle + \langle R_3\rangle + \langle C_1\rangle + \langle C_2\rangle - \langle C_3\rangle = 6.\] Each of the six terms contributes \(+1\) because the operator (including the sign in front of the term) is equal to \(+1\). However, any non-contextual theory – in the general case, not constrained by the conditions \(R_i=1\) etc. – is easily shown to obey
\[ \langle R_1\rangle + \langle R_2\rangle + \langle R_3\rangle + \langle C_1\rangle + \langle C_2\rangle - \langle C_3\rangle \leq 4.\] The proof of this inequality already emphasized that there exist Bell-like inequalities for any state that are violated in the real world. One can't be smaller or equal to four and equal to six at the same moment, so there's a clear contradiction.

Now, those experimenters use ions to measure the quantity above. Noncontextual theories predict the sum/difference of the six terms to be at most four; quantum mechanics predicts for the idealized case \(6\). Because of some experimental inaccuracies that could be described as sources "decoherence", the experimenters don't get to \(6\). They only get something like \(5.4\pm 0.05\). If they improved their gadget, they could get closer to \(6\). At any rate, the sharp and provable prediction of any non-contextual theory that you must get at most four (even when there's noise in the system) is violated at a 40-sigma level, so to say. It just doesn't hold. All non-contextual theories are dead.

Needless to say, it is redundant to perform these actual experiments because it's obvious what we get. We have tested quantum mechanics in much more accurate and complex situations than a simple two-spin measurement of an ion. Of course that if you perform the experiments accurately enough, you get an accurate enough agreement with quantum mechanics. For the sake of the argument, all these experiments could be replaced by thought experiments. For a candidate "hidden-variable theory" or another revolutionary theory to be right, the condition it has to pass is that it imitates quantum mechanics because quantum mechanics obviously does agree with all the experiments and we no longer have to test it by additional mundane low-energy experiments such as this one.

Let me say differently the reason why the \(6\) or near-\(6\) value is impossible according to non-contextual theories i.e. whenever you assume some kind of objective reality for the measurements that exists prior to the measurement. If you do, the numbers \(R_i,C_i\) are really six numbers equal to \(\pm 1\) which should "objectively exist" prior to the measurement. But because each of them may be written as the product of three \(A_{ij}\) matrix elements, we have
\[ R_1 R_2 R_3 = C_1 C_2 C_3 = \prod_{i=1}^3 \prod_{j=1}^3 A_{ij} \] The product of the three row-products is equal to the product of the three column-products: both of them are equal to the product of the nine matrix elements \(A_{ij}\) of the original matrix. So it's "logically" impossible for \(R_1R_3R_3\) to have the opposite sign than \(C_1 C_2 C_3\).

However, quantum mechanics – and the real world – doesn't respect this (classical) "logic". Two of the entries \(A_{ij}\) don't commute with each other unless they belong to the same row or the same column. In fact, for this two-spin system, they often anticommute: the product totally flips the sign if you change the order. For this reason, it's simply not true that \(\prod R_i = \prod C_i\). Instead, one may prove that \(\prod R_i = -\prod C_i\) in terms of the operators. The refusal of the observables in quantum mechanics to commute with each other is the ultimate source of the differences (and, like in this case, totally qualitative contradictions) between classical physics and quantum physics.

Fine. Imagine that an anti-quantum zealot is not a completely blinded one and he will understand that the non-contextual hidden variables are safely ruled out. The right theory of Nature has to be "contextual"; it must share a particular novel, non-classical feature of quantum mechanics known as "contextuality". Can you get anywhere?

Well, try to search for a contextual hidden-variable theory, either in your head or in the literature. You won't really find any. Such a theory would have to prepare a random result at the moment of the measurement that would depend on the type of measurements that some experimenters in the rest of the world would decide to perform. Of course that if you wanted to construct a "contextual theory with an objective reality", it would have to be non-local. Bell's theorem was more than enough to understand this point. But such a contextual theory would have to be even weirder than that; it would really have to reverse engineer the experimenters' intents and adjust all the things in such a way that the predictions of quantum mechanics are restored.

Needless to say, such a scenario is on par with a hypothetical world where God created celestial bodies, animal species, and the humans about 6,000 years ago and within a week, before He relaxed, and also digged the dinosaur fossils and all other apparent proofs of evolution into the soil. It just makes no sense. You need to deny pretty much all of the evidence to argue that such a picture is plausible. Its probability is zero for all practical purposes. The same conclusion holds for the hypothetical "contextual hidden-variable theories". Such theories would need to unnaturally adjust tons of their properties – and the only reason for (and constraint imposed upon) this adjustment is to imitate a theory (proper quantum mechanics) that can do all these things naturally; and the only reason why one would believe in such a super-contrived theory is to protect his belief in some metaphysical prejudices. Such a QM-imitating theory is not just super-unnatural and contrived; it is probably strictly impossible, too. It's more impossible than the 6,000-year-old world with dinosaur fossils.

(Some readers who find this article too technical or controversial could prefer a proof that quantum mechanics is compatible with God's glory and creationism haha.)

Some people won't give up their beliefs that there has to exist a "non-instrumentalist" underlying theory beneath quantum mechanics. Why? Because human stupidity and bigotry has no race, nationality, sex, borders, or other limits. These people just hide their heads into the sand. They will "overlook" or "instantly forget" any proof that is inconvenient for their metaphysical prejudices.

They will keep on repeating totally misleading descriptions of the reasons that make quantum mechanics different or surprising. For example, those people often say that what makes quantum mechanics different is the randomness. If one introduces some (classical) chaos to the dynamics, things will be OK.

But the fundamental difference between classical and quantum physics can't be reduced to "chaos". In the \(A_{ij}\) example above, the quantum mechanical predictions for the values of \(R_i\) and \(C_i\) were as sharp and free-of-fluctuations as the cleanest classical predictions you could think of. The "only" difference was that the sign of \(\prod(R_i C_i)\) was diametrically opposite in quantum mechanics (and reality) than it would be in any non-contextual theory.

When they start to understand that the non-determinism isn't really the only problem of their classical assumptions, they start to talk about non-locality. But non-locality isn't the real problem, either. Non-local hidden variable theories still fail. What Nature really requires us to do is to describe Her with mutually non-commuting observables. The very combination of words "non-commuting observables" says that the quantities that determine anything that may be observed and perceived simply can't be determined at the same moment.

If one uses the bizarre vocabulary of the hidden-variable apologists, he may study some of the trivial results such as the \(A_{ij}\) ion experiment above, and he may build a pile of "desired adjective" that the ultimate hidden-variable theory must satisfy. "Contextual" is one of those adjectives. However, when you combine all these adjectives and express them quantitatively, you will find out that proper quantum mechanics – with the probabilistic, instrumentalist interpretation known from the very beginning – is the only possible theory that may satisfy all these adjectives.

In this text, I argued that "contextuality" is a property of quantum mechanics; in the quantum context, it may be rephrased as the non-existence of an objective reality. However, I also want to stress that the usage of the words "contextual" and "non-contextual" is tendentious and this vocabulary is pretty much restricted to the people who are obsessed with a mission to show that quantum mechanics is fundamentally wrong or incomplete.

A person who understands that the basic framework underlying modern physics has to be quantum mechanical probably wouldn't ever talk about "contextual" things because this adjective just takes a property of quantum mechanics from the context. Of course that the values of the observables (and the predicted probabilities) depend on what you decide to measure; if you measure something that doesn't commute with \(L\), you inevitable change the probabilities of different values of \(L\).

While "contextuality" may be understood to be a valid property of quantum mechanics and our world, and I treated it in this way so far, it is a word hijacked by those who refuse quantum mechanics. Because they assume incorrect things about Nature, they're "immediately" deducing many things out of "contextuality" that are not true. Non-locality is one of them. Non-locality of the laws of physics would mean that at least in principle, events at location \(A\) change the probabilities of different things that may be seen at location \(B\). But quantum mechanics (especially its relativistic models, namely quantum field theories and string theory) denies this non-locality: locality is completely exact. The identification of "contextuality" and "non-locality" is one of the flawed operations that are made by the people who still assume that the world is intrinsically classical. They're not using quantum mechanics (and its logic) to evaluate the validity of various propositions so they're often ending with as invalid statements as \(\prod R_i = \prod C_i\) would be above.

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