Characterizing the Quantum Non-locality by the Mathematics of Magic Square

TL;DR

This paper introduces the quantum magic square as a novel mathematical framework to characterize various quantum non-local phenomena, revealing distinctive tensor structures and providing new uncertainty relations.

Contribution

It develops a high-dimensional probability tensor approach using the quantum magic square to analyze and differentiate quantum non-locality types, including Bell non-locality and entanglement.

Findings

Proves Bell and GHZ theorems within this framework

Shows uncertainty relations can quantify non-locality levels

Derives a superior conditional majorization uncertainty relation

Abstract

By constructing the quantum state in high-dimensional probability tensor, we find the quantum magic square(QMS) may stand as an ideal means of characterizing the non-local phenomena, i.e. the separability, entanglement, two/one-way steering, and Bell non-locality, etc. In this scheme, different types of non-locality exhibit distinctive inner structures of the probability tensor, which are observable in form of the partial sum of the tensor components. In application, we prove the Bell and GHZ theorems, and demonstrate that the uncertainty relation may rate the non-locality, from Bell locality to separability. We derive a conditional majorization uncertainty relation, which is superior to the steering criterion previously thought to be optimal for the uncertainty relation.