# Quantum Measurements and Contextuality

## TL;DR

This paper explores different notions of contextuality in quantum measurements, demonstrating that quantum theory is Bell noncontextual and analyzing the implications of joint probability distributions for incompatible observables.

## Contribution

It clarifies the distinction between Bell contextuality and global noncontextuality, and shows that joint probability distributions can exist without physical significance.

## Key findings

- Quantum theory is Bell noncontextual.
- Joint probability distributions can exist without physical meaning.
- The tensor product structure of quantum sample space is emphasized.

## Abstract

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, here called `globally (non)contextual' refers to whether or not there is ('noncontextual') 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.