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On Graph Approaches to Contextuality and their Role in Quantum Theory

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

In this chapter we discuss briefly the suitable mathematical structure of probabilistic models used in the description of experiments carried in a hypothetical physical system which is general enough to include Classical and Quantum Probability Theory. We discuss the notion of completion of a probabilistic model and the assumption of noncontextuality, basic ingredients for what comes in the next chapters.

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Notes

  1. 1.

    In fact, three out of these 117 measurements count twice, giving rise to a 120-vertex graph (see Fig. A.4).

  2. 2.

    To be precise, this is the definition of a projective measurement, which is not the most general measurement one can perform in a quantum system. However, repeatability suggests the restriction to projective measurements when dealing with contextuality. We will restrict our definition to this special class of measurements, and the word measurement in this book will always mean projective measurement for quantum systems. To define the most general measurement in quantum theory, one needs the notion of Generalised Measurement, related to the notion of POVM (positive operator valued measure), and the interested reader can find the definition in Ref. [NC00].

  3. 3.

    The reader familiar with the notion of hidden-variable model will notice that it is equivalent to our notion of completion. This term comes from the idea of existence of a deeper theory describing some underlying reality. However, it is important to say that such original motivation is superfluous. A completion is simply a classical probabilistic model extending the given behaviour, no matter how one interprets probabilities.

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Author information

Authors and Affiliations

  1. Department of Statistics, Physics and Mathematics, Federal University of São João del-Rei, Ouro Branco, Minas Gerais, Brazil

    Barbara Amaral

  2. International Institute of Physics, Federal University of Rio Grande do Norte, Natal, Rio Grande do Norte, Brazil

    Barbara Amaral

  3. IMECC – Department of Applied Mathematics, University of Campinas, Campinas, São Paulo, Brazil

    Marcelo Terra Cunha

Authors
  1. Barbara Amaral
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  2. Marcelo Terra Cunha
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Amaral, B., Terra Cunha, M. (2018). Introduction. In: On Graph Approaches to Contextuality and their Role in Quantum Theory. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-93827-1_1

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