Quantum Bilinear Optimization

TL;DR

This paper introduces a new hierarchy of semidefinite programming relaxations for quantum bilinear optimization problems, enabling better upper bounds on quantum advantages in various quantum information tasks.

Contribution

It presents an asymptotically converging SDP hierarchy with additional constraints, improving upon previous methods for quantum bilinear optimization problems.

Findings

Hierarchy converges asymptotically.

Additional constraints improve bounds and analytical properties.

Provides outer approximations for the completely positive semidefinite cone.

Abstract

We study optimization programs given by a bilinear form over non-commutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement-assisted coding for classical channels and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascues and Acin, our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. Moreover, this allows us to give a hierarchy of SDP outer approximations for the completely positive semidefinite cone introduced by Laurent and Piovesan.