Contextuality is a feature that arises in the empirical predictions of quantum theory, but which cannot arise in classical physics. First identified through the work of Bell  and of Kochen and Specker , it has long been studied for its foundational implications, as it crystallises key differences between the quantum and classical worlds.
If the uncertainty principle poses a fundamental obstacle to the measurement of certain combinations of observables without disturbing their respective values, it is nevertheless consistent with the possibility that such values exist “in principle” for all observables, inaccessible though they may be. The most immediate foundational consequence of contextuality is that it strips us of even this possibility.
At the same time, contextuality is increasingly investigated for its computational utility, as it has been identified as a resource that enables quantum computers to outperform classical ones: in schemes for quantum computation via magic-state distillation  or in forms of measurement-based quantum computing  for example.
Frameworks for Contextuality
A focus for contextuality research in recent times has been the development of appropriate mathematical frameworks and tools that allow us to identify, quantify, and reason about contextuality in more powerful and insightful ways. With sharper tools, we can hope to gain a deeper understanding not only of foundational experiments and results, but also of the non-classical workings of more elaborate quantum computations – knowledge that can be valuable in the development of applications for emerging quantum technologies.
It is as a result of these frameworks, for example, that it is nowadays typical to treat non-locality  as a special case of the more general phenomenon of contextuality [6,7]. While on the applied side, they have led to quantifiable relationships between contextuality and quantum-over-classical advantage in a range of informational tasks .
Topological and algebraic topological methods have proven particularly useful in this line of research. For instance, contextuality may be identified  or characterised  via topological invariants of empirical data, or via invariants of the operator arrangements abstracted from the protocols and experiments of interest [11,12]. These approaches have delivered new characterisations of the forms of contextuality that arise in measurement-based quantum computing in particular.
A homotopical toolkit
The work of Cihan Okay and Robert Raussendorf  breaks fresh ground in this line of research. Their article explores a new kind of topological abstraction whereby operator arrangements that capture the compatibility relationships between measurement operators may be represented as cell complexes. This provides another valuable perspective on contextual behaviours. Moreover, the present article blends existing approaches – which relied heavily on cohomology – with beautiful new tools from homotopy.
These developments arise as a generalisation of an earlier graph-theoretic approach due to Arkhipov , who had shown that for certain operator arrangements non-planarity in a graphical abstraction was necessary and sufficient for contextuality. Until now, that work had remained apart from the topological developments of more recent years. It also only applied in restricted scenarios. For example, operator arrangements that would typically arise in measurement-based quantum computing, and which are amenable to the cohomological approaches, lay outside of its range of applicability.
The significance of this work goes beyond providing a generalisation of Arkhipov’s result, as the rich new formalism that the authors have introduced will appeal to topologists and “contextualists” alike, and can spur further foundational and computational advances.
 JOHN S. BELL. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys., 38:447–452, Jul 1966. https://doi.org/10.1103/RevModPhys.38.447.
 E. Specker, Simon Kochen. The problem of hidden variables in quantum mechanics. Indiana Univ. Math. J., 17:59–87, 1968.
 Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the `magic' for quantum computation. Nature, 510(7505):351–355, 2014. https://doi.org/10.1038/nature13460.
 Robert Raussendorf. Contextuality in measurement-based quantum computation. Phys. Rev. A, 88:022322, Aug 2013. https://doi.org/10.1103/PhysRevA.88.022322.
 J. S. Bell. On the einstein podolsky rosen paradox. Physics Physique Fizika, 1:195–200, Nov 1964. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195.
 Samson Abramsky and Adam Brandenburger. The sheaf-theoretic structure of non-locality and contextuality. New Journal of Physics, 13(11):113036, 2011. https://doi.org/10.1088/1367-2630/13/11/113036.
 T. Fritz, A. B. Sainz, R. Augusiak, J Bohr Brask, R. Chaves, A. Leverrier, and A. Acín. Local orthogonality as a multipartite principle for quantum correlations. Nature Communications, 4(1):2263, 2013. https://doi.org/10.1038/ncomms3263.
 Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield. Contextual fraction as a measure of contextuality. Phys. Rev. Lett., 119:050504, Aug 2017. https://doi.org/10.1103/PhysRevLett.119.050504.
 Samson Abramsky, Shane Mansfield, and Rui Soares Barbosa. The cohomology of non-locality and contextuality. Electronic Proceedings in Theoretical Computer Science, 95:1–14, Oct 2012.
 Samson Abramsky, Rui Soares Barbosa, Kohei Kishida, Raymond Lal, and Shane Mansfield. Contextuality, Cohomology and Paradox. In Stephan Kreutzer, editor, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015), volume 41 of Leibniz International Proceedings in Informatics (LIPIcs), pages 211–228, Dagstuhl, Germany, 2015. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.CSL.2015.211.
 Cihan Okay, Sam Roberts, Stephen D. Bartlett, and Robert Raussendorf. Topological proofs of contextuality in quantum mechanics. Quantum Information & Computation, 17:1135–1166, 2017. https://arxiv.org/abs/1701.01888.
 Cihan Okay and Robert Raussendorf. Homotopical approach to quantum contextuality. Quantum, 4:217, January 2020. https://doi.org/10.22331/q-2020-01-05-217.
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