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arXiv · 2309.13966

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

Gereon Koßmann·René Schwonnek·Jonathan Steinberg
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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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Gereon Koßmann, René Schwonnek, Jonathan Steinberg. 2023-09-25. Hierarchies for Semidefinite Optimization in $\mathcal{C}^\star$-Algebras. https://arxiv.org/abs/2309.13966

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