# Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

## TL;DR

This paper establishes a link between Bell inequality violations and the nonexistence of symmetric extensions in qudit states, providing a new criterion for quantum state analysis based on Bell inequalities.

## Contribution

It proves that violation of certain Bell inequalities implies the nonexistence of symmetric extensions in qudit states, extending previous qubit results and proposing a test for qudit states.

## Key findings

- 2-qubit pure states do not violate CHSH inequality
- Violation of CHSH implies no symmetric extension for 2-qubit states
- Violation of monogamous Bell inequalities suggests no symmetric extension in qudits

## Abstract

We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not have a symmetric extension. For the specific case of 2-qutrit states, we use numerical evidence to conjecture that the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence, the violation of the CGLMP inequality by any 2-qutrit state could be a sufficient condition for the nonexistence of its symmetric extension.