Quantum Contextuality

Mladen Pavicic

Center of Excellence CEMS, Photonics and Quantum Optics Unit, Ruder Bošković Institute and Institute of Physics, Zagreb, Croatia

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Quantum contextual sets have been recognized as resources for universal quantum computation, quantum steering and quantum communication. Therefore, we focus on engineering the sets that support those resources and on determining their structures and properties. Such engineering and subsequent implementation rely on discrimination between statistics of measurement data of quantum states and those of their classical counterparts. The discriminators considered are inequalities defined for hypergraphs whose structure and generation are determined by their basic properties. The generation is inherently random but with the predetermined quantum probabilities of obtainable data. Two kinds of statistics of the data are defined for the hypergraphs and six kinds of inequalities. One kind of statistics, often applied in the literature, turn out to be inappropriate and two kinds of inequalities turn out not to be noncontextuality inequalities. Results are obtained by making use of universal automated algorithms which generate hypergraphs with both odd and even numbers of hyperedges in any odd and even dimensional space – in this paper, from the smallest contextual set with just three hyperedges and three vertices to arbitrarily many contextual sets in up to 8-dimensional spaces. Higher dimensions are computationally demanding although feasible.

Classical computers are binary devices while quantum ones are non-binary ones. Their discriminators are hypergraphs which determines how states supporting a computation are arranged. In quantum computers stabilizer operations initialized by superpositions of states rely on quantum gates that exhibit contextuality via contextual hypergraphs. Quantum gates are described by edges of a hypergraph.

It turns out that contextual non-binary hypergraphs are essential for designing quantum computation and communication and that their structure and implementation rely on a differentiation from their classical non-contextual binary counterparts independently of their possible coordinatization. Alternatively we can generate arbitrarily many contextual sets from simplest possible vector components and then make use of their structure by implementing the hypergraphs with the help of YES-NO measurements so as to collect data from each gate/edge and then postselect them.

This results in collecting data from the same ports/vertices belonging to different gates and eventually establish relations between vertices/vectors and edges/gates that yield several noncontextuality inequalities which serve us as alternative discriminators between contextual and noncontextual sets. The protocol consists in automated generation of hypergraphs from which contextual ones are filtered out to implement and carry out computations.

► BibTeX data

► References

[1] Ingemar Bengtsson, Kate Blanchfield, and Adán Cabello. ``A Kochen–Specker inequality from a SIC''. Phys. Lett. A 376, 374–376 (2012).

[2] Elias Amselem, Magnus Rådmark, Mohamed Bourennane, and Adán Cabello. ``State-independent quantum contextuality with single photons''. Phys. Rev. Lett. 103, 160405–1–4 (2009).

[3] B. H. Liu, Y. F. Huang, Y. X. Gong, F. W. Sun, Y. S. Zhang, C. F. Li, and G. C. Guo. ``Experimental demonstration of quantum contextuality with nonentangled photons''. Phys. Rev. A 80, 044101–1–4 (2009).

[4] Vincenzo D'Ambrosio, Isabelle Herbauts, Elias Amselem, Eleonora Nagali, Mohamed Bourennane, Fabio Sciarrino, and Adán Cabello. ``Experimental implementation of a kochen-specker set of quantum tests''. Phys. Rev. X 3, 011012–1–10 (2013).

[5] Yun-Feng Huang, Chuan-Feng Li, Yong-Sheng Zhang, Jian-Wei Pan, and Guang-Can Guo. ``Experimental test of the Kochen-Specker theorem with single photons''. Phys. Rev. Lett. 90, 250401–1–4 (2003).

[6] Gustavo Cañas, Sebastián Etcheverry, Esteban S. Gómez, C. Saavedra, Guilherme B. Xavier, Gustavo Lima, and Adán Cabello. ``Experimental implementation of an eight-dimensional Kochen-Specker set and observation of its connection with the Greenberger-Horne-Zeilinger theorem''. Phys. Rev. A 90, 012119–1–8 (2014).

[7] Gustavo Cañas, Mauricio Arias, Sebastián Etcheverry, Esteban S. Gómez, Adán Cabello, C. Saavedra, Guilherme B. Xavier, and Gustavo Lima. ``Applying the simplest Kochen-Specker set for quantum information processing''. Phys. Rev. Lett. 113, 090404–1–5 (2014).

[8] Yuji Hasegawa, Rudolf Loidl, Gerald Badurek, Matthias Baron, and Helmut Rauch. ``Quantum contextuality in a single-neutron optical experiment''. Phys. Rev. Lett. 97, 230401–1–4 (2006).

[9] H. Bartosik, J. Klepp, C. Schmitzer, S. Sponar, A. Cabello, H. Rauch, and Y. Hasegawa. ``Experimental test of quantum contextuality in neutron interferometry''. Phys. Rev. Lett. 103, 040403–1–4 (2009).

[10] G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos. ``State-independent experimental test of quantum contextuality''. Nature 460, 494–497 (2009).

[11] O. Moussa, C. A. Ryan, D. G. Cory, and R. Laflamme. ``Testing contextuality on quantum ensembles with one clean qubit''. Phys. Rev. Lett. 104, 160501–1–4 (2010).

[12] Mark Howard, Joel Wallman, Victor Veitech, and Joseph Emerson. ``Contextuality supplies the `magic' for quantum computation''. Nature 510, 351–355 (2014).

[13] Stephen D. Bartlett. ``Powered by magic''. Nature 510, 345–346 (2014).

[14] Armin Tavakoli and Roope Uola. ``Measurement incompatibility and steering are necessary and sufficient for operational contextuality''. Phys. Rev. Research 2, 013011–1–7 (2020).

[15] Debashis Saha, Paweł Horodecki, and Marcin Pawłowski. ``State independent contextuality advances one-way communication''. New J. Phys. 21, 093057–1–32 (2019).

[16] Claude Berge. ``Graphs and hypergraphs''. Volume 6 of North-Holland Mathematical Library. North-Holland. Amsterdam (1973).

[17] Claude Berge. ``Hypergraphs: Combinatorics of finite sets''. Volume 45 of North-Holland Mathematical Library. North-Holland. Amsterdam (1989).

[18] Alain Bretto. ``Hypergraph theory: An introduction''. Springer. Heidelberg (2013).

[19] Vitaly I. Voloshin. ``Introduction to graph and hypergraph theory''. Nova Science. New York (2009).

[20] Simon Kochen and Ernst P. Specker. ``The problem of hidden variables in quantum mechanics''. J. Math. Mech. 17, 59–87 (1967). url: http:/​/​www.jstor.org/​stable/​24902153.

[21] Adán Cabello. ``Experimentally testable state-independent quantum contextuality''. Phys. Rev. Lett. 101, 210401–1–4 (2008).

[22] Piotr Badziág, Ingemar Bengtsson, Adán Cabello, and Itamar Pitowsky. ``Universality of state-independent violation of correlation inequalities for noncontextual theories''. Phys. Rev. Lett. 103, 050401–1–4 (2009).

[23] Asher Peres. ``Two simple proofs of the Bell-Kochen-Specker theorem''. J. Phys. A 24, L175–L178 (1991).

[24] Michel Planat and Metod Saniga. ``Five-qubit contextuality, noise-like distribution of distances between maximal bases and finite geometry''. Phys. Lett. A 376, 3485–3490 (2012).

[25] Karl Svozil and Josef Tkadlec. ``Greechie diagrams, nonexistence of measures and Kochen–Specker-type constructions''. J. Math. Phys. 37, 5380–5401 (1996).

[26] Karl Svozil. ``Quantum logic''. Discrete Mathematics and Theoretical Computer Science. Springer-Verlag. New York (1998).

[27] Karl Svozil. ``New forms of quantum value indefiniteness suggest that incompatible views on contexts are epistemic''. Entropy 20, 535–541 (2018).

[28] Adán Cabello, José R. Portillo, Alberto Solís, and Karl Svozil. ``Minimal true-implies-false and true-implies-true sets of propositions in noncontextual hidden-variable theories''. Phys. Rev. A 98, 012106–1–8 (2018).

[29] Karl Svozil. ``What is so special about quantum clicks?''. Entropy 22, 1–43 (2020).

[30] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson. ``Kochen-specker contextuality''. Rev. Mod. Phys. 94, 0450007–1–62 (2022). arXiv:2102.13036.

[31] M. Planat. ``On small proofs of the Bell-Kochen-Specker theorem for two, three and four qubits''. Eur. Phys. J. Plus 127, 86–1–11 (2012).

[32] Mordecai Waegell and P. K. Aravind. ``Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions''. J. Phys. A 44, 505303–1–15 (2011).

[33] Mladen Pavičić, Jean-Pierre Merlet, Brendan D. McKay, and Norman D. Megill. ``Kochen-Specker vectors''. J. Phys. A 38, 1577–1592 (2005).

[34] Mladen Pavičić, Jean-Pierre Merlet, Brendan D. McKay, and Norman D. Megill. ``CORRIGENDUM Kochen-Specker vectors''. J. Phys. A 38, 3709 (2005).

[35] Sixia Yu and C. H. Oh. ``State-independent proof of Kochen-Specker theorem with 13 rays''. Phys. Rev. Lett. 108, 030402–1–5 (2012).

[36] Petr Lisoněk, Piotr Badzi¸ag, José R. Portillo, and Adán Cabello. ``Kochen-Specker set with seven contexts''. Phys. Rev. A 89, 042101–1–7 (2014).

[37] Adán Cabello, Elias Amselem, Kate Blanchfield, Mohamed Bourennane, and Ingemar Bengtsson. ``Proposed experiments of qutrit state-independent contextuality and two-qutrit contextuality-based nonlocality''. Phys. Rev. A 85, 032108–1–4 (2012).

[38] Zhen-Peng Xu, Jing-Ling Chen, and Hong-Yi Su. ``State-independent contextuality sets for a qutrit''. Phys. Lett. A 379, 1868–1870 (2015).

[39] Ravishankar Ramanathan and Pawel Horodecki. ``Necessary and sufficient condition for state-independent contextual measurement scenarios''. Phys. Rev. Lett. 112, 040404–1–5 (2014).

[40] Adán Cabello, Matthias Kleinmann, and Costantino Budroni. ``Necessary and sufficient condition for quantum state-independent contextuality''. Phys. Rev. Lett. 114, 250402–1–5 (2014).

[41] Mladen Pavičić. ``Hypergraph contextuality''. Entropy 21(11), 1107–1–20 (2019).

[42] Xiao-Dong Yu and D. M. Tong. ``Coexistence of Kochen-Specker inequalities and noncontextuality inequalities''. Phys. Rev. A 89, 010101(R)–1–4 (2014).

[43] Xiao-Dong Yu, Yan-Qing Guo, and D. M. Tong. ``A proof of the Kochen–Specker theorem can always be converted to a state-independent noncontextuality inequality''. New J. Phys. 17, 093001–1–7 (2015).

[44] Asher Peres. ``Incompatible results of quantum measurements''. Phys. Lett. A 151, 107–108 (1990).

[45] N. David Mermin. ``Simple unified form for the major no-hidden-variable theorem''. Phys. Rev. Lett. 65, 3373–3376 (1990).

[46] Mladen Pavičić and Norman D. Megill. ``Automated generation of arbitrarily many Kochen-Specker and other contextual sets in odd dimensional Hilbert spaces''. Phys. Rev. A 106, L060203–1–5 (2022).

[47] Adán Cabello, Matthias Kleinmann, and José R. Portillo. ``Quantum state-independent contextuality requires 13 rays''. J. Phys. A 49, 38LT01–1–8 (2016).

[48] Asher Peres. ``Quantum theory: Concepts and methods''. Kluwer. Dordrecht (1993).

[49] Michael Kernaghan. ``Bell-Kochen-Specker theorem for 20 vectors''. J. Phys. A 27, L829–L830 (1994).

[50] Adán Cabello, José M. Estebaranz, and Guillermo García-Alcaine. ``Bell-Kochen-Specker theorem: A proof with 18 vectors''. Phys. Lett. A 212, 183–187 (1996).

[51] Mladen Pavičić. ``Kochen-Specker algorithms for qunits'' (2004). arXiv:quant-ph/​041219.

[52] Mladen Pavičić, Norman D. Megill, and Jean-Pierre Merlet. ``New Kochen-Specker sets in four dimensions''. Phys. Lett. A 374, 2122–2128 (2010).

[53] Mladen Pavičić. ``Vector generation of quantum contextual sets: QTech2018, Paris, video'' (January 2019). https:/​/​www.youtube.com/​watch?v=Bw2vItz5trE.

[54] Adán Cabello, Simone Severini, and Andreas Winter. ``Graph-theoretic approach to quantum correlations''. Phys. Rev. Lett. 112, 040401–1–5 (2014).

[55] Barbara Amaral and Marcelo Terra Cunha. ``On graph approaches to contextuality and their role in quantum theory''. SBMAC Springer. (2018).

[56] Mladen Pavičić, Brendan D. McKay, Norman D. Megill, and Krešimir Fresl. ``Graph approach to quantum systems''. J. Math. Phys. 51, 102103–1–31 (2010).

[57] Norman D. Megill and Mladen Pavičić. ``Kochen-Specker sets and generalized Orthoarguesian equations''. Ann. Henri Poinc. 12, 1417–1429 (2011).

[58] Mladen Pavičić. ``Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets''. Phys. Rev. A 95, 062121–1–25 (2017).

[59] Mladen Pavičić and Norman D. Megill. ``Vector generation of quantum contextual sets in even dimensional Hilbert spaces''. Entropy 20, 928–1–12 (2018).

[60] Mladen Pavičić, Mordecai Waegel, Norman D. Megill, and P.K. Aravind. ``Automated generation of Kochen-Specker sets''. Scientific Reports 9, 6765–1–11 (2019).

[61] Mordecai Waegell and P. K. Aravind. ``Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem''. J. Phys. A 43, 105304–1–13 (2010).

[62] Mordecai Waegell and P. K. Aravind. ``Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group''. Phys. Rev. A 88, 012102–1–10 (2013).

[63] Mordecai Waegell and P. K. Aravind. ``Parity proofs of the Kochen-Specker theorem based on 120-cell''. Found. Phys. 44, 1085–1095 (2014).

[64] Mordecai Waegell and P. K. Aravind. ``Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8''. J. Phys. A 48, 225301–1–17 (2015).

[65] Mordecai Waegell, P. K. Aravind, Norman D. Megill, and Mladen Pavičić. ``Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell''. Found. Phys. 41, 883–904 (2011).

[66] Richard J. Greechie. ``Orthomodular lattices admitting no states''. J. Comb. Theory A 10, 119–132 (1971).

[67] Gudrun Kalmbach. ``Orthomodular logic''. Z. math. Logik Grundl. Math. 20, 395–406 (1974).

[68] Karl Svozil. ``Extensions of Hardy-type true-implies-false gadgets to classically obtain indistinguishability''. Phys. Rev. A 103, 022204–1–13 (2021).

[69] Adán Cabello. ``Converting contextuality into nonlocality''. Phys. Rev. Lett. 127, 070401–1–7 (2021).

[70] Karl Svozil. ``Generalized Greenberger–Horne–Zeilinger arguments from quantum logical analysis''. Found. Phys. 52, 4–1–23 (2022).

[71] Adán Cabello. ``Twin inequality for fully contextual quantum correlations''. Phys. Rev. A 87, 010104(R)–1–5 (2013).

[72] Jason Zimba and Roger Penrose. ``On Bell non-locality without probabilities: More curious geometry''. Stud. Hist. Phil. Sci. 24, 697–720 (1993).

[73] Arthur Fine and Paul Teller. ``Algebraic constraints on hidden variables''. Found. Phys. 8, 629–636 (1978).

[74] Mordecai Waegell and P. K. Aravind. ``Parity proofs of the Kochen-Specker theorem based on 24 rays of Peres''. Found. Phys. 41, 1785–1799 (2011).

[75] John S. Bell. ``On the problem of hidden variables in quantum mechanics''. Rev. Mod. Phys. 38, 447–452 (1966).

[76] A. M. Gleason. ``Measures on the closed subspaces of a Hilbert space''. J. Math. Mech. 6, 885–893 (1957). url: http:/​/​www.jstor.org/​stable/​24900629.

[77] Karl-Peter Marzlin and Taylor Landry. ``On the connection between the theorems of Gleason and of Kochen and Specker''. Can. J. Phys. 93, 1446–1452 (2015).

[78] Alexander A. Klyachko, M. Ali Can, Sinem Binicioğlu, and Alexander S. Shumovsky. ``Simple test for hidden variables in spin-1 systems''. Phys. Rev. Lett. 101, 020403–1–4 (2008).

[79] Adán Cabello. ``Simple explanation of the quantum violation of a fundamental inequality''. Phys. Rev. Lett. 110, 060402–1–5 (2013).

[80] Piotr Badziág, Ingemar Bengtsson, Adán Cabello, Helena Granström, and Jan-Åke Larsson. ``Pentagrams and paradoxes''. Found. Phys. 41, 414–423 (2011).

[81] Arthur R. Swift and Ron Wright. ``Generalized Stern-Gerlach experiments and the observability of arbitrary spin operators''. J. Math. Phys. 21, 77–82 (1980).

[82] C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan. ``State-independent experimental test of quantum contextuality in an indivisible system''. Phys. Rev. Lett. 109, 150401–1–5 (2012).

[83] M. Grötschel, L. Lovász, and A. Schrijver. ``The ellipsoid method and its consequences in combinatorial optimization''. Combinatorica 1, 169–197 (1981).

[84] O. Melnikov, V. Sarvanov, R. Tysbkevich, V. Yemelichev, and I. Zverovich. ``Exercises in graph theory''. Kluwer. Dordrecht (1998).

[85] Karol Horodecki, Jingfang Zhou, Maciej Stankiewicz, Roberto Salazar, Paweł Horodecki, Robert Raussendorf, Ryszard Horodecki, Ravishankar Ramanathan, and Emily Tyhurst. ``The rank of contextuality''. arXiv (2022).

[86] Andrzej Dudek, Joanna Polcyn, and Andrzej Ruciński. ``Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence''. J. Comb. Optim. 19, 184–199 (2010).

[87] Richard P. Feynman, Robert B. Leighton, and Mathew Sands. ``The Feynman lectures on physics; Volume III. Quantum mechanics''. Addison-Wesley. Reading, Massachusetts (1965). url: https:/​/​www.feynmanlectures.caltech.edu/​.

[88] Julio T. Barreiro, Tzu-Chieh Wei, and Paul G. Kwiat. ``Beating the channel capacity limit for linear photonic superdense coding''. Nature Phys. 4, 282–286 (2008).

[89] Julio T. Barreiro, Tzu-Chieh Wei, and Paul G. Kwiat. ``Remote preparation of single-photon ``hybrid'' entangled and vector-polarization states''. Phys. Rev. Lett. 105, 030407–1–4 (2010).

[90] Mladen Pavičić, Norman D. Megill, and Jean-Pierre Merlet. ``New Kochen-Specker sets in four dimensions''. Phys. Lett. A 374, 2122–2128 (2010).

[91] Mladen Pavičić and Norman D. Megill. ``Vector generation of contextual sets''. EPJ Web of Conferences 198, 00009 (2019) 198, 00009–1–8 (2019).

[92] Jeffrey Bub. ``Schütte's tautology and the Kochen-Specker theorem''. Found. Phys. 26, 787–806 (1996).

[93] Jan-Åke Larsson. ``A Kochen-Specker inequality''. Europhys. Lett. 58, 799–805 (2002).

[94] Carsten Held. ``Kochen-specker theorem''. In D. Greenberger, K. Hentschel, and F. Weinert, editors, Compendium of Quantum Physics. Pages 331–335. Springer, New-York (2009).

[95] N. David Mermin. ``Hidden variables and the two theorems of John Bell''. Rev. Mod. Phys. 65, 803–815 (1993).

[96] Roger Penrose. ``On Bell non-locality without probabilities: Some curious geometry''. In John Ellis and Daniele Amati, editors, Quantum Reflections. Pages 1–27. Cambridge University Press, Cambridge (2000).

[97] Andrés Cassinello and Antonio Gallego. ``The quantum mechanical picture of the world''. Am. J. Phys. 73, 273–281 (2005).

[98] Mladen Pavičić. ``Companion to quantum computation and communication''. Wiley-VCH. Weinheim (2013).

[99] Mladen Pavičić, Norman D. Megill, P. K. Aravind, and Mordecai Waegell. ``New class of 4-dim Kochen-Specker sets''. J. Math. Phys. 52, 022104–1–9 (2011).

[100] Ali Asadian, Costantino Budroni, Frank E. S. Steinhoff, Peter Rabl, and Otfried Gühne. ``Contextuality in phase space''. Phys. Rev. Lett. 114, 250403–1–5 (2020).

[101] Adán Cabello, José M. Estebaranz, and Guillermo García-Alcaine. ``Recursive proof of the Bell-Kochen-Specker theorem in any dimension $n>3$''. Phys. Lett. A 339, 425–429 (2005).

[102] Mordecai Waegell and P. K. Aravind. ``Minimal complexity of Kochen-Specker sets does not scale with dimension''. Phys. Rev. A 95, 050101 (2017).

[103] Tycho Sleator and Harald Weinfurter. ``Realizable universal quantum logic gates''. Phys. Rev. Lett. 74, 4087–4090 (1995).

[104] P. Kurzyński and D. Kaszlikowski. ``Contextuality of almost all qutrit states can be revealed with nine observables''. Phys. Rev. A 86, 042125–1–4 (2012).

[105] Pawel Kurzyński, Adán Cabello, and Dagomir Kaszlikowski. ``Fundamental monogamy relation between contextuality and nonlocality''. Phys. Rev. Lett. 112, 100401–1–5 (2014).

[106] G'abor Hofer-Szabó. ``Three noncontextual hidden variable models for the Peres-Mermin square''. Euro. J. Phil. Sci. 11, 1–12 (2021).

Cited by

[1] Mladen Pavičić and Norman D. Megill, "Automated generation of arbitrarily many Kochen-Specker and other contextual sets in odd-dimensional Hilbert spaces", Physical Review A 106 6, L060203 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2023-03-23 10:36:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-03-23 10:36:58).