GAMP/QMATH Lecture: Semidefinite Optimization for Quantum Information Processing
Semidefinite Optimization for Quantum Information Processing
Abstract: We apply semidefinite programming to study two basic lines of quantum information processing: entanglement manipulation and communication over quantum channels. For entanglement manipulation, we resolved a longstanding open problem by showing that positive-partial-transpose-preserving (PPT) operations do not lead to a reversible entanglement theory. This means that even we relax the free operations from LOCC operations to PPT operations, the asymptotic transformation between quantum states is still irreversible. In the framework of communication, we introduce semidefinite programs (SDPs) for estimating the coding rate and success probability for classical communication over quantum channels. We establish the first general SDP upper bound on the classical capacity of a quantum channel and give the best known upper bound for the classical capacity of the amplitude damping channel. We also establish a finite resource analysis of communication over quantum erasure channels, including the first second-order expansion of classical capacity beyond entanglement-breaking channels.
Based on the works arXiv:1606.09421, arXiv:1610.06381, arXiv:1709.052