# 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.