Measures of contextuality

Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Although it was recognized before Bell's nonlocality, despite some recent proposals, it still faces a fundamental problem: how to quantify its presence ? In this work, we introduce two measures of contextuality. One is direct analogue of a known measure of non-locality, called the contextuality cost. The other can be viewed as an analogue of the relative entropy of entanglement and is called the relative entropy of contextuality. Based on the fundamental fact that contextual system can not be described by a single joint probability distribution, we introduce another measure of contextuality, called optimal contextual correlation factor, and prove that it equals relative entropy of contextuality, providing thereby operational derivation of the latter. We further show that it is monotonous under some of operations which preserve non-contextuality. We provide a lower bound on relative entropy of contextuality by computing its variant called uniform relative entropy of contextuality, for the boxes that possess high symmetries including Popescu-Rohrlich, Peres-Mermin, Mermin's and Klyachko's ones. In special cases we prove the additivity of uniform relative entropy of contextuality, while in others show that it is additive in two copies. We compute also the cost of contextuality of some boxes, and observe that it does not increase under operations which preserve non-contextuality.