Abstract
Recent years have seen new general notions of contextuality emerge. Most of these employ context-independent symbols to represent random variables in different contexts. As an example, the operational theory of (Spekkens in Phys Rev A 71(5):52108, 2005) treats an observable being measured in two different contexts identically. Non-contextuality in this approach is the impossibility of drawing ontological distinctions between identical elements of the operational theory. However, a recent collection of work seeks to exploit context-dependent symbols of random variables to interpret contextuality (Kujala et al. in Phys Rev Lett 115(15):150401, 2015; Dzhafarov and Kujala in Phys Scr T163:014009, 2014). This approach associates contextuality with the possibility of imposing a particular joint distribution on random variables recorded under different experimental contexts. This paper compares these two different treatments of random variables and highlights the limitations of the context-dependent approach as a physical theory.
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Notes
For simplicity, from now on we will use \(a_{i}^{j}\) instead of \(a_{q_{i}}^{c_{j}}\), where subscripts i indicate different observables and j indicate different contexts.
It is clear that one can check whether the two random variables \(a_{\mathbb {A}_{1}}^{(\mathbb {A}_{1}, \mathbb {B}_{1})}\) and \(a_{\mathbb {A}_{1}}^{(\mathbb {A}_{1}, \mathbb {B}_{2})}\) have the same distribution only when they have a same value.
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Aliakbarzadeh, M., Kitto, K. Is contextuality about the identity of random variables?. Found Phys 51, 14 (2021). https://doi.org/10.1007/s10701-021-00402-7
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DOI: https://doi.org/10.1007/s10701-021-00402-7