# Genuine monogamy relations in no-signaling theories - a geometric approach

## TL;DR

This paper introduces a geometric approach to derive genuine monogamy relations in no-signaling theories, revealing that only one Bell inequality out of many can be violated in multipartite quantum systems, and extends these results to qudits.

## Contribution

It presents a novel geometric method to establish genuine monogamy relations for Bell inequalities in multipartite and high-dimensional systems, including qubits and qudits.

## Key findings

- Derived genuine monogamy relations for multipartite Bell inequalities.
- Showed only one Bell inequality can be violated among many in these systems.
- Extended the approach to qudits and tripartite systems, deriving new Bell inequalities.

## Abstract

Quantum correlations are subject to certain distribution rules represented by so-called monogamy relations. Derivation of monogamy relations for multipartite systems is a non-trivial problem, as the multipartite correlations reveal their behaviors in a way different from bipartite systems. We here show that simple geometric properties of probabilistic spaces, in conjunction with no-signaling principle, lead to genuine monogamy relations for a large class of Bell type inequalities for many qubits. The term of 'genuine' implies that only one out of $N$ Bell inequalities exhibits a quantum violation. We also generalize our method to qudits. Using the similar geometric approach with a quasi-distance employed, we derive Svetlichny-Zohren-Gill type Bell inequalities for $d$-dimensional tripartite systems, and show their monogamous nature.