Title: Sum-of-squares and extendibility hierarchies from copositive matrices
Speaker: Sang-Jun Park（Wuhan University）
Time: Wednesday, June 10, 2026, 10:00-11:30
Venue：Gewu Building 315
Abstract:
Extendibility hierarchies of quantum states, originating from the seminal work of Doherty, Parrilo, and Spedalieri, provide efficient yet complete families of entanglement criteria. The dual notion of extendibility is also remarkably connected to sum-of-squares (SOS) hierarchies for certain classes of complex polynomials. In this talk, we address two closely related questions: (1) Can one construct quantum states that are entangled while exhibiting a high degree of extendibility? (2) Do there exist entanglement witnesses that do not admit an SOS certificate at any level of the hierarchy? We answer both questions by exploiting the structure of copositive matrices, which have been extensively studied in polynomial optimization.
First, we show that the SOS hierarchies used to approximate the cone of copositive matrices can be naturally lifted to the dual cones of entanglement witnesses associated with the PPT (positive-partial-transpose) bosonic extendibility hierarchy. In particular, we construct a large family of entanglement witnesses that cannot be certified at any level of the PPT bosonic extendibility hierarchy, thereby answering a long-standing open question posed by Doherty, Parrilo, and Spedalieri. Using this duality, we further provide an explicit construction of bipartite diagonal symmetric states that are simultaneously entangled and PPT-bosonic-exchangeable for any desired hierarchy level r.
This is joint work with Aabhas Gulati and Ion Nechita.
