Contextuality without incompatibility?

[Vaxjo]

TL;DR

Let us compare the fundamental notions of quantum physics - contextuality vs. incompatibility.
The natural question arises: What is contextuality without incompatibility? (What is ``dry-residue''?)

see https://link.springer.com/article/10.1007/s10773-020-04666-z
I think that contextuality is just another word for incompatibility. I asked may people: Point out to the dry-residue of contextuality, filtered of incompatibility. All these bright people were not able to say anything sensible on this issue. They continue to highlight contextuality as some distinguishing feature of quantum physics.
P.S. I speak about quantum physics, quantum observables, not mathematics with generalized probability models.
I would be happy to receive some input on this issue.
Yours, Andrei Khrennikov

 

[DrChinese]

I think that contextuality is just another word for incompatibility.

I could not agree with this statement. Conceivably, there could be contextuality without what you call "incompatibility"; and vice versa. In the area of QM: we know from Bell that models are contextual. And we know that the contextual observables are also incompatible (where incompatible means non-commuting in the sense of the Heisenberg uncertainty relations. It might even be true that observables are contextual if and only if they are incompatible. But that does not make them the same thing.

More importantly, I am not sure if your statement is useful. People argue over whether contextuality is the same thing as "no hidden variables" and whether quantum non-locality requires contextuality. These ideas are all related in discussions of QM, but some of this simply evolves into semantics with little science involved.

 

[Vaxjo]

Thanks a lot for your comment! It seems that we agree on one very important point: there is no contextuality without incompatibility! Good! But, nevertheless, is there something additional in contextuality, so to say complementary to incompatibility? My point is that nothing. In my paper attached to the post I derived some mathematical condition, characterizing contextuality's component complementary to incompatibility. However, I was not able to find any physical meaning behind this condition.

P.S. "More importantly, I am not sure if your statement is useful. People argue over whether contextuality is the same thing as "no hidden variables" and whether quantum non-locality requires contextuality. These ideas are all related in discussions of QM, but some of this simply evolves into semantics with little science involved."
Well, do you want to say that contextuality= incompatibility, and difference is only in words? For me, this is a good point for discussion: Have we done something new comparing with Bohr's studies on complementarity? I am not sure.

 

[Sunil]

I think that contextuality is just another word for incompatibility.

I disagree.

We measure A together with B1 or together with B2. The measurement of A is contextual - the result depends on what was measured, B1 or B2. But there is no incompatibility related with A. Measuring A is compatible as with measuring B1, as with measuring B2.

There is some incompatibility, but it is not directly related to A - it is the incompatibility of measuring B1 together with B2.

 

[Vaxjo]

Thanks a lot for your contribution to this discussion. You presented the standard definition of contextuality. Heuristically it is very attractive. It has just one casualty: it has no relation to physics, since it is based on COUNTERFACTUAL reasoning. This is not physics, but metaphysics. Such statements could not be verified experimentally. A few years ago, Karl Svozil claimed that he designed an experimental test to check this type of contextuality, but I am very sceptical w.r.t. to it.
K. Svozil, Proposed direct test of a certain type of noncontextuality in quantum mechanics. Phys. Rev. A 80, 04010 (2009).

 

[Demystifier]

@Vaxjo I have an example of contextuality without incompatibility.

Consider a repeated projective measurement of spin of a single particle. Suppose that I first measure spin in the z-direction and that I get +1/2. If then I repeat the measurement in the z-direction an arbitrary number of times, I will always get the same result, +1/2. But after a few repetitions, suppose that I measure in the x-direction and obtain some result (not important which). And suppose that then I measure in the z-direction again. Now there is a 50% chance that I will get the opposite result, -1/2. Clearly, the measurement in the x-direction influenced spin in the z-direction. That's a manifestation of contextuality. Yet no incompatibility is involved.

 

[Vaxjo]

Well, from my viewpoint this is just incompatibility of spin observables corresponding to z and x -axes. I suspect that Bohr and Heisenberg would reply you in this way.

 

[Sunil]

You presented the standard definition of contextuality. Heuristically it is very attractive. It has just one casualty: it has no relation to physics, since it is based on COUNTERFACTUAL reasoning. This is not physics, but metaphysics. Such statements could not be verified experimentally.

Once based on this "metaphysics" one can derive theorems about observables, like Kochen-Specker, it is as physical as everything in physical theories.

 

[Vaxjo]

No, Kochen-Specker theorem is derived from existence of incompatible observables, this is its essence. If you like let us concretely discuss K-S theorem. What is additional to the existence of incompatible observables? Suppose Bohr or Heisenberg would read about this theorem, would they be excited? I am sure that not.

 

[Demystifier]

Well, from my viewpoint this is just incompatibility of spin observables corresponding to z and x -axes. I suspect that Bohr and Heisenberg would reply you in this way.

No. Incompatibility means that you cannot know/measure/define/whatever both at the same time. But in my example, they are measured at different times, which is not forbidden. Anyway, the notion of contextuality did not even exist at the time of Bohr and Heisenberg, so I'm not sure how they would call it.

 

[Demystifier]

Anyway, the notion of contextuality did not even exist at the time of Bohr and Heisenberg, so I'm not sure how they would call it.

@Vaxjo why do you always escape from your threads when it becomes interesting? Anyway, it just occurred to me that, even though Bohr did not use the word "contextuality", he often stressed that one must take the whole experimental setup into account. That's precisely what contextuality is!