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Bounding the separable rank via polynomial optimization
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NSTL
Elsevier
We investigate questions related to the set SEPd consisting of the linear maps rho acting on Cd circle times Cd that can be written as a convex combination of rank one matrices of the form xx*circle times 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(rho), defined as the smallest number of rank one states xx* circle times yy* entering the decomposition of a separable state rho. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter tau sep(rho), a natural convexification of the combinatorial parameter ranksep(rho). A distinguishing feature is exploiting the positivity constraint rho - xx* circle times 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,
NSTL
Elsevier
