Kochen-Specker theoreom (1967) established the notion of contextuality which is usually termed as KS-contextuality. Spekken (2005) generalizes the definition of contextuality. Spekken introduces three components of contextuality: preparation, transformation, and measurement. Here I am interested in measurement contextuality. He claims that his notion of measurement contextuality cleanly separates the notion of outcome determinism. That is, if we add the notion of outcome determinism to Spekken's measurement contextuality, we will get KS-contextuality.
Whenever I read it long enough I feel that I understand it but unable to construct any concrete example. Peres-Mermin square provides a simple example to illustrate the notion of KS-contextuality, but a similar and concrete example isn't available for measurement contextuality in which system isn't outcome deterministic. Yes, Spekken's paper has a proof but it is not illuminating that give a feel of it.
Let me explain my confusion in a plain manner. Nontextuality assumption is that measurement procedures in the same equivalence class generates same probability distributions of measurement outcomes. However if a measurement procedures in the same equivalence class generate a different probability distribution of measurement outcomes, then this violates noncontextuality assumption, hence an example of measurement contextuality.
My confusion is that first we require an equivalence class which itself defined by same probability distribution, how can then we have pick a measurement procedure from the the equivalance class that generates a different probability distribution. We want to identify the features which are not captured by the equivalence class.
The set of compatible observables, say A, defines an equivalence class of observables. All these observables will give same probability distribution of measurement results. The probability distribution may change if I choose an incompatible observable with respect to the set A. But this observable would not be in the same equivalence class as the observable A belong to.
Perhaps I should think as follows: A measurement noncontextual (physical) system has only a single context. That is, no measurement procedure to test a specific property would yield different probability distributions of measurement results. If any two measurements generates two different probability distribution of measurement outcomes then the system is measurement contextual.
Please correct me if this line of thinking is incorrect or missing some point. A simple example, like Peres-Mermin square, will be greatly helpful.
