Quantum contextuality is a foundational concept in quantum mechanics stating that the outcome one gets in a measurement is dependent upon what other measurements one is trying to make. More formally, the measurement result of a quantum observable is dependent upon which other commuting observables are within the same measurement set.
Contextuality was first proposed in the Bell-Kochen-Specker theorem, which revealed that non-contextual hidden variable theories are incompatible with quantum mechanics. That is, the state of a quantum system cannot be described either deterministically or independent of the experimental setup. Since then, contextuality has developed under several mathematical frameworks, including the Sheaf Theoretic, Spekkens' operational contextuality, and the graph theoretic. The Sheaf Theoretic proposed by Samson Abramsky and Adam Brandenburger employs sheaf theory to generalize contextuality to all forms of measurement, not just measurements in quantum mechanics. Meanwhile, Spekkens defines and expands upon contextuality as it applies to quantum information and experimentation, and the graph theoretic explains contextuality using the mathematical formalism present in graph theory. There also exists a theory known as "contextuality by default". This theory recently inspired researchers to apply the concept of contextuality to human psychology and sociology and has met with mixed results.
Recently, quantum contextuality has been explored as a potential tool for quantum computing. In 2013, Robert Raussendorf showed that in general, any Mermin-like, inequality-free proof for quantum contextuality (such as the Kochen-Specker theorem) can be turned into a measurement-based quantum computation. More formally, "under quite natural assumptions for multi-qubit systems, MBQCs (measurement based quantum computations) which compute a non-linear Boolean function with sufficiently high success probability are contextual." Then in 2014, Mark Howard, et. al. showed that applying contextuality to magic state distillation (MSD) provides a means for arriving at universal, fault-tolerant quantum computing. MSD is a process by which a single qubit's polarization is increased along one of several "magic" directions and its state is considered "magic" once its polarization reaches a certain level. In particular, the group proved that states are non-contextual if and only if they cannot be used as inputs for magic state distillation. This provides a selection criteria for proper input states needed in MSD, and by extension aids in the efficient creation of magic states that, when coupled with fault-tolerant operators, allows for universal quantum computing. In both cases, contextuality has a potential advantage over other quantum computational techniques since contextuality itself can be thought of as a theory of information that is built directly into quantum mechanics. Comparatively, more traditional quantum phenomena such as entanglement and interference use delicate physical states which are incredibly susceptible to noise and very difficult to manipulate experimentally.
Kochen and Specker
Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that quantum mechanics is contextual for systems of dimension 3 and greater. The Kochen-Specker Theorem demonstrates a contradiction between two foundational assumptions in hidden the hidden variable theories for explaining quantum mechanics, that quantum-mechanical observables have definite values and that these values are independent of the system making a measurement of said observable. This latter assumption is an articulation of noncontextuality in quantum mechanics. In addition, Kochen and Specker constructed an explicitly noncontextual hidden variable model for the two-dimensional qubit case in their paper on the subject., thereby completing the characterisation of the dimensionality of quantum systems that can demonstrate contextual behaviour. Bell's proof invoked a weaker version of Gleason's theorem, reinterpreting the result to show that quantum contextuality exists only in dimensions greater than two. Note that this is the dimensionality of the Hilbert space, not of space-time.
Graph theory and optimization
Adán Cabello, Simone Severini, and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories. This allowed them to show that quantum contextuality is closely related to the Lovász number, an important parameter used in optimization and information theory. By making use of similar techniques, Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson have shown that the Lovász number has a key role in determining the power of quantum computing.
The notion of quantum contextuality due to Spekkens removes the assumption of determinism of outcomes that is present in other forms of contextuality. This breaks the interpretation of contextuality as a direct extension of Nonlocality, and does not treat irreducible randomness as nonclassical. However, such forms of contextuality can be motivated using Leibniz's law of the Identity of indiscernibles; the law applied to physical systems mirrors the modified definition of noncontextuality. This was further explored by Simmons et al, who demonstrated that other notions of contextuality could also be motivated by Leibnizian principles, and could be thought of as tools enabling ontological conclusions from operational statistics.
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- Abramsky, Samson; Brandenburger, Adam (2011-11-28). "The Sheaf-Theoretic Structure Of Non-Locality and Contextuality". New Journal of Physics. 13 (11): 113036. arXiv:1102.0264. doi:10.1088/1367-2630/13/11/113036. ISSN 1367-2630.
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- Cabello, Adan; Severini, Simone; Winter, Andreas (2014-01-27). "Graph-Theoretic Approach to Quantum Correlations". Physical Review Letters. 112 (4): 040401. arXiv:1401.7081. doi:10.1103/PhysRevLett.112.040401. ISSN 0031-9007. PMID 24580419.
- Dzhafarov, Ehtibar N.; Kujala, Janne V.; Cervantes, Victor H. (2015-04-02). "Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology". arXiv:1504.00530. doi:10.1007/978-3-319-28675-4-2 (inactive 2018-11-30).
- Kujala, Janne; Zhang, Ru; Dzhafarov, Ehtibar (2015-04-28). "Is there contextuality in behavioral and social systems?". doi:10.1098/rsta.2015.0099.
- Cervantes, Victor H.; Dzhafarov, Ehtibar N. (2018-11-30). "True Contextuality in a Psychophysical Experiment". arXiv:1812.00105 [quant-ph, q-bio].
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- Raussendorf, Robert (2013-08-19). "Contextuality in Measurement-based Quantum Computation". Physical Review A. 88 (2). arXiv:0907.5449. doi:10.1103/PhysRevA.88.022322. ISSN 1050-2947.
- Howard, Mark; Wallman, Joel; Veitch, Victor; Emerson, Joseph (2014-06). "Contextuality supplies the 'magic' for quantum computation". Nature. 510 (7505): 351–355. arXiv:1401.4174. doi:10.1038/nature13460. ISSN 0028-0836. PMID 24919152. Check date values in:
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- A. Cabello, S. Severini, A. Winter, Graph-Theoretic Approach to Quantum Correlations", Physical Review Letters 112 (2014) 040401.
- R. Spekkens, "Contextuality for preparations, transformations, and unsharp measurements",Phys. Rev. A 71, 052108 (2005).
- A.W. Simmons, Joel J. Wallman, H. Pashayan, S.D. Bartlett, T. Rudolph, "Contextuality under weak assumptions", New J. Phys. 19 033030, (2017).