Varieties of contextuality based on probability and structural nonembeddability

TL;DR

The paper differentiates probabilistic and strong notions of quantum contextuality and uses Kochen-Specker’s embeddability criterion to separate them. Through hull-based Boole-Bell analyses, functional gadgets, and explicit KS-type configurations, it shows how quantum probabilities can violate classical bounds while some structures still admit classical embeddings, whereas others do not. It emphasizes value indefiniteness and partial value functions as foundational concepts, arguing that strong contextuality implies a fundamental scarcity or absence of classical two-valued states. The discussion links these theoretical distinctions to historical developments and experimental interpretations, underscoring the philosophical and computational implications for quantum foundations.

Abstract

Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's Theorem~0 is a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic "strong" form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed "value indefinite" and formalized by partial functions of theoretical computer sciences.

Figures (3)

• Figure 1: (a) Configuration of 13 observables in seven bi-intertwining contexts serving as a symmetric (with respect to horizontal) true-implies-false gadget; (b) the associated classical probability distributions obtained by the convex sum $\lambda_1+\cdots +\lambda_{14}=1$, $\lambda_i\ge 0$, $1 \le i \le 14$; (c) a quantum representation in terms of a faithful orthogonal vertex labeling in terms of a vertex representation by vectors, preserving orthogonality of adjacent vertices lovasz-79 of the hypergraph that maximizes the probability of $a_7$, given $a_1$.
• Figure 2: The Kochen-Specker "combo of Specker bugs" whose set of classical truth assignments formalized by its two-valued states cannot separate $a_1$ from $b_1$, as well as $a_7$ from $b_7$.
• Figure 3: A configuration of quantum observables with a nonunital set of classical two-valued states in three-dimensional Hilbert space Svozil-2018-p. Admissibility demands that proposition $a_1$ must be true (value 1); and the adjacent propositions $a_2$, $a_{13}$, $a_{15}$, $a_{16}$, $a_{17}$, $a_{25}$, $a_{27}$, $a_{36}$ sharing hyperedges with $a_1$ must be false (value 0). All other observables are either 0 or 1, depending on the respective two-valued state enumerated in Table \ref{['2021-context-table-ACS']}.