# Bell non-locality and Kochen-Specker contextuality: How are they connected?

## TL;DR

This paper explores the complex relationships between Bell non-locality and Kochen-Specker contextuality in quantum theory, establishing a comprehensive map of their connections and transformations.

## Contribution

It introduces a theory-independent framework and constructs a detailed map linking Bell non-locality and KS contextuality, including transformation methods between their correlation matrices.

## Key findings

- Every Bell non-local correlation matrix can be mapped to a KS contextual correlation matrix.
- Partial and one-to-one connections exist between KS and Bell correlations in certain scenarios.
- Any KS correlation matrix in dimension d can be transformed into a Bell correlation matrix between two d-dimensional systems.

## Abstract

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.