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I work as a theorist with my current research interests in "quantum contextuality". For those perusing the question, this is essentially a generalization of non-locality where we can show a quantum system does not admit a hidden-variable model. So in some sense, it's impossible to look at the problem classically.

For further background, contextuality can often be formulated as a resource for quantum computation-- which is the reason why we're interested in it. Our talking points to people are that contextuality is the right way of characterizing what's essentially quantum about a system (many would debate us on that, but that's perhaps for another thread).

What I've found so far is a proposed experiment for testing contextuality: Cabello-2016-Experiment and a description of the implications of a successful contextuality experiment: Winter-2014-Implications.

Most experimentalists tend to focus on entanglement (understandably; it's a lot easier to measure and more concrete to think about) - but are there any experimental groups currently working on contextuality experiments?

Sanchayan Dutta
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Emily Tyhurst
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    Might be relevant also PhysRev A paper. People in queens from that paper I believe have ties to experiments but whether specifically on contextuality experiments I don't know. – snulty Mar 15 '18 at 00:42
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    Curiosity, but is "quantum contextuality" generally held to be a uniquely quantum phenomena, or is it meant as an analog to some sort of non-quantum contextuality? – Nat Mar 15 '18 at 00:50
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    I wanted to clarify that I'm interested in contextuality as it applies to quantum systems, however, some classical systems have been known to exhibit contextuality, e.g. arXiv:1310.4990. In general resource-wise, we are interested in state-dependent contextuality, for which I don't think classical systems exhibit. – Emily Tyhurst Mar 15 '18 at 02:57

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I don't think I can answer this question precisely, but I would like to say some things.

Since this question was asked, much has changed in the field. We have the following known sources that may interest any reader passing by (since probably by now the OP is an expert in the subject, I hope this answer can be useful):

With all that in mind I just want to make some small comments:

There are experimental implementations but no arguably loophole-free

It is a matter of great debate what notion of contextuality is optimal, moreover, every notion has its peculiarities and the search is to a broad notion that is experimentally robust and without too many hidden assumptions. Many experiments have been performed and I don't know any groups that are currently performing experiments, but I know that every group searches for an experimental implementation of contextuality that is arguably loophole-free. As far as I know, probably, groups are working with experimental data from experiments performed in Latin America and USA/Europe trying to grasp CbD from data tables. Groups are trying to perform experiments in quantum contextuality to apply in random number generation. Groups are proposing experimental implementations of violations of contextuality/nonlocality inequalities using quantum linear optics. There probably are groups trying to implement new loophole-free experimental tests of generalized contextuality (but I only know about the theoretical developments). Since this is an ongoing active research field, different groups do not easily share their experiments much before having good enough results.

Loopholes

As there is not a specific notion of contextuality that is considered the best (and likely there will never be, I think it more possible that all different notions will be better applied in different circumstances) different notions have different loopholes to worry about. In generalized contextuality, for example, an important loophole is the fact that ideal operational equivalences are usually not present in real-life experiments, and secondary procedures must be described (see the experimental references I have mentioned already). See for example this presentation here discussing loophole-free tests in the framework of Abramsky-Brandenburger contextuality.

Is "quantum contextuality" generally held to be a uniquely quantum phenomenon, or is it meant as an analog to some sort of non-quantum contextuality?

Contextuality is not a quantum phenomenon. Contextuality is a phenomenon that is satisfied by quantum theory. This is something that is made clear by every description of contextuality, in particular, almost every presentation by Samson Abramsky has the maxima that contextuality refers to the fact that probabilistic data may present

"local consistency but global inconsistency"

and to me, this description makes clear that there is nothing quantum about contextuality, but there is something contextual about quantum! And we know this for more than 50 years now, that quantum theory cannot be explained by contextual empirical models (sticking to sheaf-approach terminology in this sentence at least).

When we describe the notion of Generalized Contextuality, we treat it in the framework of operational-probabilistic theories, or also in the GPT framework, which has quantum theory as a particular instance.

In the CbD approach, this is also noticed as the fact that contextuality is a property of random variables, and not of quantum theory itself. The most relevant aspect of this discussion corresponds to the fact that exist post-quantum correlations.

Adding to the discussion in the comment section: we know that every noncontextual empirical model has a quantum realization, and with that I want to say that given any scenario, the probabilities arising in this scenario (depends on the scenario, non-local scenarios for example have probabilities of the form $p(ab|xy)$ with $a,b$ possible results for $x,y$ measurements in each party side) have a description in terms of the Born rule (therefore, $p(ab|xy) = Tr(\rho E_a^x \otimes E_b^y)$). With contextuality not being a quantum aspect, I mean that we don't need quantum theory to infer if a given data-table is contextual or not. Quantum theory is a specific type of theory that cannot be explained only by noncontextual data-tables. Less known is that this property is true both for KS contextuality and for Generalized contextuality as well.

R.W
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  • thanks for the great answer! When you say "Contextuality is not a quantum phenomenon. Contextuality is a phenomenon that is satisfied by quantum theory." do you mean that there are probabilistic classical scenarios which are contextual? That would be quite interesting. Do you have a source or simple example of this? (if the answer is nontrivial, I can ask this as a separate question) – glS Jul 06 '21 at 14:56
  • No, I don't mean this. In this field, classicality is what we understand as noncontextuality, so every data-table that is classical is defined to be every noncontextual data-table. Every noncontextual data-table can be described by quantum theory. So, there are probabilistic nonclassical data-tables, in known scenarios, that provide us with non-classical (hence contextual) and also non-quantum empirical models. I have inserted the PR box original paper as the first place where this came about. – R.W Jul 06 '21 at 15:10
  • But I think your comment is so relevant that I should add a small note in the answer. Thanks a lot for the commentary. Att: RW – R.W Jul 06 '21 at 15:11
  • when you say "data-tables", do you mean sets of conditional probabilities ${p(a|x)}_{a,x}$ (where $a,x$ are possibly multiple input/outputs)? That is, what in other contexts are called behaviours? – glS Jul 06 '21 at 15:27
  • so am I correct in saying that any "classical" (as in, explainable with local hidden variables) probability distribution, is noncontextual? In your edit you are saying something much stronger though. If I'm reading this correctly, you say that you can always explain behaviours (i.e. conditional probability distributions) via QM. That sounds too strong: isn't this in contrast with the fact that there are behaviours, e.g. superluminal ones, which cannot be reproduced via QM? – glS Jul 06 '21 at 15:33
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    " I correct in saying that any "classical" (as in, explainable with local hidden variables) probability distribution, is noncontextual?" Yes, Noncontextuality implies locality (in multipartite scenarios). "you say that you can always explain behaviours (i.e. conditional probability distributions) via QM." I think not, I say that ever noncontextual behaviour is also quantum behaviour. "isn't in contrast with the fact that there are behaviours, e.g. superluminal ones." In the noncontextuality scenarios framework, non-signaling is always an assumption made. You mean superluminal NC models? – R.W Jul 06 '21 at 15:54
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    One can consider superluminal effects as possible effects in causal descriptions such as M-contextuality or the theory of causal-inference. But what I know is that we do have proofs regarding the fact that every noncontextual behaviour is a quantum behaviour. The question regarding superluminal noncontextual models is, I think, well-suited for a post question on its own, so that the discussion can be useful for other users. I also think that classicality/noncontextuality/quantumness of sets is another interesting post question on its own. – R.W Jul 06 '21 at 15:57
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    Just to mention, behaviours and data-tables and empirical models are all different names for the (almost) same things. – R.W Jul 06 '21 at 15:58
  • if there is an implicit no-signalling assumption, that answers my doubt, thanks – glS Jul 06 '21 at 15:58
  • In (almost) every non-contextuality scenario it is assumed that behaviours are non-disturbing. In multipartite scenarios this becomes the well-known non-signalling conditions. CbD approach is one that allows for disturbance. But this is a hole complication on its own. Another question for another post probably. Att. Rafa – R.W Jul 06 '21 at 16:00
  • Thank you a lot for all your comments, suggestions, and questions. – R.W Jul 06 '21 at 16:01
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The Quantum Reality group at the Centre for Quantum Technologies (National University of Singapore) https://qreality.quantumlah.org/

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