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

摘要

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