Kochen-Specker contextuality

Abstract

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics is in conflict with classical models in which the result of a measurement does not depend on which other compatible measurements are jointly performed. Here compatible measurements are those that can be implemented simultaneously or, more generally, those that are jointly measurable. This conflict is generically called quantum contextuality. In this review, an introduction to this subject and its current status is presented. Several proofs of the Kochen-Specker theorem and different notions of contextuality are reviewed. How to experimentally test some of these notions is explained, and connections between contextuality and nonlocality or graph theory are discussed. Finally, some applications of contextuality in quantum information processing are reviewed.

Partial Bell-State Measurement with Type-II Parametric Down Conversion: Extracting Phase Information from the Zeropoint Field (I)

“…In this sense, in Equations ( 83 ), ( 84 ), ( 86 ), ( 87 ), ( 93 ) and ( 94 ), the maximum and minimum values of the SIAMs in the rectilinear and diagonal analyses, respectively, show the following counterintuitive results: (a) the corresponding values of the maximal SIAMs are different; (b) the values of the corresponding intensities to non activated detectors are different in both basis; (c) the minimal values of the SIAMs in the rectilinear (diagonal) analysis are positive (negative), which can be deduced by substituting the relationship between (see Equation ( 74 )) into Equations ( 93 ) and ( 94 ). From our point of view, these results are consistent with the quantum contextuality [ 32 , 33 , 34 , 35 , 36 ], but this subject deserves further study.…”

Quantum Contextuality

“…hypergraph is called a "simplified version" of a graph and an MMP hyperedge is said to "correspond to a clique of size 6" but the term MMP hyperedges is neither mentioned nor cited. Budroni, Cabello, Gühne, Kleinmann and Larsson [30] make use of both terms, graphs and hypergraphs, interchangeably. They state that "contexts can be represented as graphs, or more generally hypergraphs" [30, p. 30].…”

“…• Greechie diagrams have recently been used as a name for what are actually MMP hypergraphs [27,28,29,30]. This is a misnomer since Greechie diagrams-connected Hasse diagrams-belong to the field of partially ordered sets and can represent neither graphs, nor general hypergraphs, nor MMP hypergraphs.…”

“…For instance, for vertices 1,2,5,10,15,18, i.e., edges 1-10, 2-15 and 5-18, the probabilities p 1,10 , p 2,15 , and p 5,18 were obtained. However, as shown in [93,33,34,94,58,41,30] not just two but four vertices should be measured and should have a coordinatization in each green and red hyperedge even when we do not take all of them into account while postprocessing the data.…”

“…2. Historically, representations used to describe and depict contextual sets appeared in several different forms and definitions, e.g., partial Boolean algebra [20], operator and projectors [21,22], lists or tables of vectors and their orthogonalities [23,24], Greechie diagrams [25,26,27,28,29,30], Kochen-Specker (KS) proofs [31], parity proofs [32], MMP diagrams [33,34], graphs with cliques [35], nodecontext graphs [36,30], etc. However, as shown in this paper, when some discrepancies between these definitions as well as their possible inner limitations are smoothed out, all of them boil down to hypergraphs and in this paper we provide a hypergraph platform for major results and achievements in the field of quantum contextual sets.…”