Sang-Jun Park: [WS1] Sum-of-squares methods in quantum entanglement theory

Speaker
Sang-Yung Park, Wuhan University

Abstract
Extendibility hierarchies of quantum states, originating from the seminal work of Doherty, Parrilo, and Spedalieri, provide efficient yet complete families of entanglement criteria. Moreover, the dual notion of extendibility has a remarkable connection with sum-of-squares (SOS) hierarchies for certain classes of complex polynomials. In particular, this connection shows how the sum-of-squares method provides a powerful framework for studying optimization problems arising in quantum entanglement theory. The first part of the talk will introduce the basic notions and address two closely related questions: (1) Can we construct quantum states that are PPT (positive-partial-transpose) entangled yet exhibit a high degree of extendibility? (2) Are there entanglement witnesses that do not admit an SOS certificate at any level of the hierarchy? In the second part, based on recent joint work with Aabhas Gulati and Ion Nechita, we answer both questions by establishing precise connections between the entanglement properties of classes of symmetric states with diagonal unitary symmetry and SOS hierarchies for copositive matrices.