Gadget structures in proofs of the Kochen-Specker theorem

TL;DR

This paper introduces the concept of $01$-gadgets within Kochen-Specker graphs, revealing their fundamental role in constructing proofs of the Kochen-Specker theorem and related quantum contextuality arguments.

Contribution

It identifies $01$-gadgets as essential substructures in Kochen-Specker graphs, providing a new primitive for proofs and arguments in quantum foundations.

Findings

Every Kochen-Specker graph contains a $01$-gadget.

$01$-gadgets can be used to construct various Kochen-Specker proofs.

The approach simplifies and unifies 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.