Quantum Measurements and Contextuality

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'.