Gadget structures in proofs of the Kochen-Specker theorem

Ravishankar Ramanathan1, Monika Rosicka2, Karol Horodecki3,4, Stefano Pironio5, Michał Horodecki2,6, and Paweł Horodecki6,7

1Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
2Institute of Theoretical Physics and Astrophysics and the National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-308 Gdansk, Poland.
3Institute of Informatics Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-308 Gdansk, Poland
4International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland
5Laboratoire d'Information Quantique, Université Libre de Bruxelles, Belgium
6International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
7Faculty of Applied Physics and Mathematics, National Quantum Information Centre, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland

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The Kochen-Specker theorem is a fundamental result in quantum foundations that has spawned massive interest since its inception. We show that within every Kochen-Specker graph, there exist interesting subgraphs which we term $01$-gadgets, that capture the essential contradiction necessary to prove the Kochen-Specker theorem, i.e,. every Kochen-Specker graph contains a $01$-gadget and from every $01$-gadget one can construct a proof of the Kochen-Specker theorem. Moreover, we show that the $01$-gadgets form a fundamental primitive that can be used to formulate state-independent and state-dependent statistical Kochen-Specker arguments as well as to give simple constructive proofs of an ``extended'' Kochen-Specker theorem first considered by Pitowsky in [22].

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► References

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Cited by

[1] Yuan Liu, Ravishankar Ramanathan, Karol Horodecki, Monika Rosicka, and Paweł Horodecki, "Optimal measurement structures for contextuality applications", npj Quantum Information 9 1, 63 (2023).

[2] Shuai Zhao, Ravishankar Ramanathan, Yuan Liu, and Paweł Horodecki, "Tilted Hardy paradoxes for device-independent randomness extraction", Quantum 7, 1114 (2023).

[3] Karl Svozil, "Extensions of Hardy-type true-implies-false gadgets to classically obtain indistinguishability", Physical Review A 103 2, 022204 (2021).

[4] Václav Voráček and Mirko Navara, "Generalised Kochen–Specker Theorem in Three Dimensions", Foundations of Physics 51 3, 67 (2021).

[5] Ravishankar Ramanathan, Yuan Liu, and Paweł Horodecki, "Large violations in Kochen Specker contextuality and their applications", New Journal of Physics 24 3, 033035 (2022).

[6] Adán Cabello, "Converting Contextuality into Nonlocality", Physical Review Letters 127 7, 070401 (2021).

[7] Mohammad H. Shekarriz and Karl Svozil, "Noncontextual coloring of orthogonality hypergraphs", Journal of Mathematical Physics 63 3, 032104 (2022).

[8] Mordecai Waegell and P. K. Aravind, "Golay codes and quantum contextuality", Physical Review A 106 6, 062421 (2022).

[9] Karl Svozil, "What Is So Special about Quantum Clicks?", Entropy 22 6, 602 (2020).

[10] Ravishankar Ramanathan, Michał Horodecki, Hammad Anwer, Stefano Pironio, Karol Horodecki, Marcus Grünfeld, Sadiq Muhammad, Mohamed Bourennane, and Paweł Horodecki, "Practical No-Signalling proof Randomness Amplification using Hardy paradoxes and its experimental implementation", arXiv:1810.11648, (2018).

[11] Karl Svozil, "Classical predictions for intertwined quantum observables are contingent and thus inconclusive", arXiv:1808.00813, (2018).

[12] Karl Svozil, "Varieties of contextuality based on probability and structural nonembeddability", arXiv:2103.06110, (2021).

[13] Mladen Pavicic and Mordecai Waegell, "Generation of Kochen-Specker contextual sets in higher dimensions by dimensional upscaling whose complexity does not scale with dimension and their applications", arXiv:2305.08267, (2023).

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