Logical Reflections on the Kochen-Specker Theorem

Ravi Kunjwal

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.

Andrew Lenard

Asher Peres

Victoria J Wright, Ravi Kunjwal

The Kochen–Specker (KS) theorem reveals the nonclassicality of single quantum systems. In contrast, Bell's theorem and entanglement concern the nonclassicality of composite quantum systems. Accordingly, unlike incompatibility, entanglement and Bell non-locality are not necessary to demonstrate KS-contextuality. However, here we find that for multiqubit systems, entanglement and non-locality are both essential to proofs of the Kochen–Specker theorem. Firstly, we show that unentangled measurements (a strict superset of local measurements) can never yield a logical (state-independent) proof of the KS theorem for multiqubit systems. In particular, unentangled but nonlocal measurements—whose eigenstates exhibit ''nonlocality without entanglement"—are insufficient for such proofs. This also implies that proving Gleason's theorem on a multiqubit system necessarily requires entangled projections, as shown by Wallach [Contemp Math, 305: 291-298 (2002)]. Secondly, we show that a multiqubit state admits a statistical (state-dependent) proof of the KS theorem if and only if it can violate a Bell inequality with projective measurements. We also establish the relationship between entanglement and the theorems of Kochen–Specker and Gleason more generally in multiqudit systems by constructing new examples of KS sets. Finally, we discuss how our results shed new light on the role of multiqubit contextuality as a resource within the paradigm of quantum computation with state injection.

Roderick I. Sutherland

Eliza Miller

There is ongoing discourse in philosophical and physics literature about whether the probabilistic essence of quantum mechanics indicates that deterministic interpretations of nature must be false. In 1967 two mathematicians, Simon Kochen and Ernst Specker, developed a mathematical theorem to the effect that particles do not have well-defined property values at a given time. A logical consequence of this is that there can be no well-defined hidden variables responsible for the outcomes of mysterious quantum behaviors. The theorem is thus fatal to the many deterministic interpretations of quantum mechanics that assume hidden variables, including pilot wave mechanics, a quantum theory developed by Louis de Broglie and David Bohm a few decades prior to the formulation of the Kochen-Specker Theorem. The purpose of this essay is to evaluate the legitimacy of the logical basis of Kochen’s and Specker’s argument considering its severe implications. It is concluded that Kochen and Specker do not provide sufficient evidence that deterministic theories must fail, due to their misinterpretation of the axioms on which hidden variable theories rely. This paper also finds that pilot wave mechanics is mathematically successful while remaining consistent with the deterministic understanding of broader physics and deserving of more serious consideration as a quantum theory. Above all it is emphasized that the nonsensicality of absolute randomness and indeterminism should be acknowledged, and theorems purporting to prove their existence should be critically examined, for such a discovery would undermine the legitimacy of logic itself.