Papers › Bounding the separable rank via polynomial optimization
Bounding the separable rank via polynomial optimization
Sander Gribling, Monique Laurent, Andries Steenkamp
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We investigate questions related to the set 𝒮ℰ𝒫_d consisting of the linear maps ρ acting 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 𝒮ℰ𝒫_d and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from (A.C. Doherty, P.A. Parrilo and F.M. Spedalieri. Distinguishing separable and entangled states. Phys. Rev. Lett. 88(18):187904, 2002).
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