Bounding the separable rank via polynomial optimization
Gribling,Sander Laurent,Monique Steenkamp,Andries
Gribling,Sander
Laurent,Monique
Steenkamp,Andries
Abstract
We investigate questions related to the set SEPd consisting of the linear maps ρacting on Cd⊗Cd that can be written as a convex combination of rank one matrices of the form xx∗⊗yy∗. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank ranksep(ρ), defined as the smallest number of rank one states xx∗⊗yy∗ entering the decomposition of a separable state ρ. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter τsep(ρ), a natural convexification of the combinatorial parameter ranksep(ρ). A distinguishing feature is exploiting the positivity constraint ρ −xx∗⊗yy∗ 0 to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set SEPd and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from Doherty et al. (2002) [16].
Description
Publisher Copyright: © 2022
Date
2022-09-01
Journal Title
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Volume Title
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Research Projects
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Keywords
Approximation hierarchies, Completely positive rank, Entanglement, Matrix factorization ranks, Polynomial optimization, Separable rank
Citation
Gribling, S, Laurent, M & Steenkamp, A 2022, 'Bounding the separable rank via polynomial optimization', Linear Algebra and its Applications, vol. 648, pp. 1-55. https://doi.org/10.1016/j.laa.2022.04.010