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

TL;DR

This paper develops a unified framework for semidefinite relaxations of cone programs within $\

Contribution

It introduces a general approach to relaxations of cone programs on $\

Findings

Connects hierarchies like NPA and Lasserre to $\

Shows structural similarities between different hierarchies in $\

Provides a general perspective on symmetry reductions in SDPs

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.