Minimal true-implies-false and true-implies-true sets of propositions in noncontextual hidden variable theories

Abstract

An essential ingredient in many examples of the conflict between quantum theory and noncontextual hidden variables (e.g., the proof of the Kochen-Specker theorem and Hardy's proof of Bell's theorem) is a set of atomic propositions about the outcomes of ideal measurements such that, when outcome noncontextuality is assumed, if proposition $A$ is true, then, due to exclusiveness and completeness, a nonexclusive proposition $B$ ($C$) must be false (true). We call such a set a {\em true-implies-false set} (TIFS) [{\em true-implies-true set} (TITS)]. Here we identify all the minimal TIFSs and TITSs in every dimension $d \ge 3$, i.e., the sets of each type having the smallest number of propositions. These sets are important because each of them leads to a proof of impossibility of noncontextual hidden variables and corresponds to a simple situation with quantum vs classical advantage. Moreover, the methods developed to identify them may be helpful to solve some open problems regarding minimal Kochen-Specker sets.