Hierarchies for Semidefinite Optimization in ${\mathcal{C}}^{\star }$-Algebras

Abstract

Semidefinite Optimization has become a standard technique in the landscape of Mathematical Programming that has many applications in finite dimensional Quantum Information Theory. This paper presents a way for finite-dimensional relaxations of general cone programs on ${\mathcal{C}}^{\star }$-algebras which have structurally similar properties to ordinary cone programs, only putting the notion of positivity at the core of optimization. We show that well-known hierarchies for generalized problems like NPA but also Lasserre's hierarchy and to some extend symmetry reductions of generic SDPs by de-Klerk et al. can be considered from a general point of view of ${\mathcal{C}}^{\star }$-algebras in combination to optimization problems.