Bell non-locality and Kochen-Specker (KS) contextuality are logically
independent concepts, fuel different protocols with quantum vs classical
advantage, and have distinct classical simulation costs. A natural question is
what are the relations between these concepts, advantages, and costs. To
address this question, it is useful to have a map that captures all the
connections between Bell non-locality and KS contextuality in quantum theory.
The aim of this work is to introduce such a map. After defining the
theory-independent notions of Bell non-locality and KS contextuality for ideal
measurements, we show that, in quantum theory, due to Neumark's dilation
theorem, every matrix of quantum Bell non-local correlations can be mapped to
an identical matrix of KS contextual correlations produced in a scenario with
identical relations of compatibility but where measurements are ideal and no
space-like separation is required. A more difficult problem is identifying
connections in the opposite direction. We show that there are "one-to-one" and
partial connections between KS contextual correlations and Bell non-local
correlations for some KS contextuality scenarios, but not for all of them.
However, there is also a method that transforms any matrix of KS contextual
correlations for quantum systems of dimension d into a matrix of Bell
non-local correlations between two quantum subsystems each of them of dimension
d. We collect all these connections in map and list some problems which can
benefit from this map.Comment: 13 pages, 2 figure
