Gadget structures in proofs of the Kochen-Specker theorem

Abstract

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.

arXiv comment19 pages, 8 figures, Accepted in Quantum

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• Roots and (re)sources of value (in)definiteness versus contextuality. A contribution to the Pitowsky Volume in memory of Itamar Pitowsky (1950--2010)
• Large violations in Kochen Specker contextuality and their applications
• Noncontextual coloring of orthogonality hypergraphs
• The Golay codes and Quantum Contextuality
• Varieties of contextuality based on probability and structural nonembeddability
• Optimal Measurement Structures for Contextuality Applications
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• Generation of Kochen-Specker contextual sets in higher dimensions by dimensional upscaling whose complexity does not scale with dimension and their applications
• Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets