Quantum Measurements and Contextuality. (arXiv:1902.05633v1 [quant-ph])

In quantum physics the term `contextual' can be used in more than one way.
One usage, here called `Bell contextual' since the idea goes back to Bell, is
that if $A$, $B$ and $C$ are three quantum observables, with $A$ compatible
(i.e., commuting) with $B$ and also with $C$, whereas $B$ and $C$ are
incompatible, a measurement of $A$ might yield a different result (indicating
that quantum mechanics is contextual) depending upon whether $A$ is measured
along with $B$ (the $\{A,B\}$ context) or with $C$ (the $\{A,C\}$ context). An
analysis of what projective quantum measurements measure shows that quantum
theory is Bell noncontextual: the outcome of a particular $A$ measurement when
$A$ is measured along with $B$ would have been exactly the same if $A$ had,
instead, been measured along with $C$.

A different definition found in Samson Abramsky et. al., Phys. Rev. Lett.,
119:050504, 2017, here called `globally (non)contextual' to distinguish it from
the Bell variety, refers to whether there is (`noncontextual'), or is not
(`contextual'), an 'empirical model' that simultaneously assigns probabilities
in a consistent manner to the outcomes of measurements of a certain collection
of observables, not all of which are compatible. A simple example shows that an
empirical model can exist even in a situation where the (supposed) measurement
probabilities cannot refer to properties of a quantum system, and hence lack
physical significance, even though mathematically well-defined. It is noted
that the quantum sample space required for interpreting measurements of
incompatible properties in separate runs of an experiment has a tensor product
structure, a fact sometimes overlooked.

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