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

• URL: http://arxiv.org/abs/2309.13966v1
• Date: Mon, 25 Sep 2023 09:01:30 GMT
• Title: Hierarchies for Semidefinite Optimization in $\mathcal{C}^\star$-Algebras
• Authors: Gereon Ko{\ss}mann, Ren\'e Schwonnek and Jonathan Steinberg
• Abstract summary: This paper presents a way for finite-dimensional relaxations of general cone programs on $mathcalCstar$-algebras. 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.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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.

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