Abstract
Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framework to obtain noise-robust witnesses of contextuality from such scenarios. Our approach builds on the results of R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015), by providing new insights into the relationship between the structure of a contextuality scenario and the associated noise-robust noncontextuality inequalities that witness contextuality. The present work also forms a necessary counterpart to the framework presented in R. Kunjwal, Quantum 3, 184 (2019), which only applies to KS-colourable contextuality scenarios, i.e., those which do not admit logical proofs of the KS theorem but do admit statistical proofs. We rely on a single hypergraph invariant, defined in R. Kunjwal, Quantum 3, 184 (2019), that appears in our contextuality witnesses, namely, the weighted max-predictability. The present work can also be viewed as a study of this invariant. Significantly, unlike the case of R. Kunjwal, Quantum 3, 184 (2019), none of the graph invariants from the graph-theoretic framework for KS-contextuality due to Cabello, Severini, and Winter (the ``CSW framework", Phys. Rev. Lett. 112, 040401 (2014)) are relevant for our noise-robust noncontextuality inequalities.
Highlights
Our approach builds on the results of R
We have presented a framework for noise-robust noncontextuality inequalities that are inspired by logical proofs of the Kochen-Specker theorem
We have identified special sets of these inequalities, corresponding to irreducible minimally indeterministic sets of contexts, that are independent of each other and can generate any other noise-robust noncontextuality inequality corresponding to a minimally indeterministic set of contexts (MISC) or even any other set of contexts
Summary
The Kochen-Specker (KS) theorem [1] stands out as a fundamental insight into the nature of quantum measurements, formalizing the fact that these measurements cannot always be understood as merely revealing pre-existing values of physical quantities. Recent work [6,7,8,9,10,11] has taken the first steps towards turning the insight of the Kochen-Specker theorem into operational constraints — or noncontextuality inequalities — that are robust to noise and experimentally testable These inequalities do not presume the structure of quantum measurements — in particular, that they are projective — in their derivation, relying only on operational constraints that can be verified in an experiment and make sense, in particular, for nonprojective measurements in quantum theory.. The approach we develop in this paper allows one to identify the physical quantities that one can expect to be constrained by the assumption of noncontextuality (and why) in KS-uncolourable contextuality scenarios This is in contrast to the approach adopted in Refs.
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