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 non-contextual: 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, here called ‘globally (non)contextual’ refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution 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 such a joint probability distribution can exist even in a situation where the 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, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked.
This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.