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

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.

6 pages, 3 figures. Version 2: Corrected typos, added a reference, added a paragraph at the end of proof of theorem 1

References in corpus (5)

• A complete family of separability criteria
• Detecting multipartite entanglement
• Symmetric Extension of Two-Qubit States
• Spectrum conditions for symmetric extendible states
• Symmetric extendibility for qudits and tolerable error rates in quantum cryptography

Cited by in corpus (2)

• Extendibility of bosonic Gaussian states
• Eigenstate entanglement in integrable collective spin models