Analytical techniques on multilinear problems

Resumo

This Ph.D. dissertation mainly focuses on three multilinear problems and itsaimistodescribe analytical and topological techniques that we found useful to tackle these problems. The first problem comes from Quantum Information theory, it is the so-called the Separability Problem, and the other two were proposed by Gurariy. Let Mk denote the set of complex matrices of order k and let Pk be the set of positive semidefinite Hermitian matrices of Mk. The aim of this problem is to find a deterministic criterion to distinguish the separable states from the entangled states. In this work we shall only deal with the bipartite finite dimensional case, therefore the states are elements in the tensor product space Mk ?Mm. We say that B ? Mk ?Mm is separable if B =?in=1 Ci ?Di, where Ci ? Pk and Di ? Pm, for every i. If B is not separable then B is entangled. Denote by VMkV the set {V XV,X ? Mk}, where V ? Mk is an orthogonal projection. We say that a linear transformation T :VMkV ?WMmW is a positive map, if T(Pk ?VMkV )? Pm ?WMmW. We say that a non null positive map T : VMkV ?VMkV is irreducible if V ? MkV ? ? VMkV is such that T(V ? MkV ?)? V ? MkV ? then V ? = V or V ? = 0. Let us say that T : VMkV ? VMkV is a completely reducible map, if it is a positive map and if there are orthogonal projections V1,...,Vs ? Mk such that ViVj = 0 (i ? j), ViV = Vi (1 ? i ? s), VMkV = V1MkV1 ? ... ? VsMkVs ? R, R ? V1MkV1 ? ... ? VsMkVs satisfying: T(ViMkVi)? ViMkVi (1 ? i ? s), TSis irreducible (1 ? i ? s), TSR ? 0. Let A =?ni=1 Ai ?Bi ? Mk ?Mm. Define GA : Mk ?Mm, as GA(X)=?ni=1 tr(AiX)Bi and FA : Mm ? Mk, as FA(X)=?in=1 tr(BiX)Ai. Our main results are the following: If A ? Mk ?Mm is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC) or invariant under realignment then FA ?GA : Mk ? Mk is completely reducible...