Técnicas analíticas en problemas multilineales

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, leída el 16/09/2016This 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 =Σin=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 =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Define GA : Mk →Mm, as GA(X)=Σni=1 tr(AiX)Bi and FA : Mm → Mk, as FA(X)=Σin=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...Esta tesis doctoral se centra principalmente en tres problemas multilineales y su objetivo es describir las técnicas analíticas y topológicas útiles para atacar estos problemas. El primer problema tiene su origen en la Teoría de Información Cuántica, es el llamado problema de la separabilidad de los estados cuánticos, y los otros dos fueron propuestos por Vladimir I. Gurariy. Denotemos por Mk al conjunto de las matrices complejas de orden k y Pk será el conjunto de matrices Hermíticas semidefinidas positivasde Mk. El objetivo de nuestro primer problema es encontrar un criterio determinístico para distinguir los estados separables de los estados entrelazados. Aqui sólo trabajamos con el caso bipartito de dimensión finita, luego los estados son los elementos del producto tensorial Mk ⊗Mm. Decimos que B ∈ Mk ⊗Mm es separable si B =Σni=1 Ci ⊗Di, donde Ci ∈ Pk y Di ∈ Pm, para cada i. Si B no es separable entonces B está entrelazada. Sea VMkV el conjunto {V XV,X ∈ Mk}, donde V ∈ Mk es una proyección ortogonal. Se dice que una transformación lineal T : VMkV →WMmW es una aplicación positiva, si T(Pk ∩VMkV )⊂ Pm ∩WMmW. Se dice que una aplicación no nula positiva T : VMkV → VMkV es irreducible si V ′ MkV ′ ⊂ VMkV es tal que T(V ′ MkV ′)⊂ V ′ MkV ′ entonces V ′ = V o V ′ = 0. Digamos que T : VMkV → VMkV es una aplicación completamente reducible, si es positiva y si hay proyecciones ortogonales V1,...,Vs ∈ Mk tales que ViVj = 0 (i ≠ j), ViV = Vi (1 ≤ i ≤ s), VMkV = V1MkV1⊕...⊕VsMkVs⊕R, R ⊥V1MkV1⊕...⊕VsMkVs y que satisfacen: T(ViMkVi)⊂ ViMkVi (1 ≤ i ≤ s), TSViMkVi es irreducible (1 ≤ i ≤ s), TSR ≡ 0. Sea A =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Defina GA : Mk →Mm, GA(X)=1 tr(AiX)Bi y FA : Mm →Mk, FA(X)=Σin=1 tr(BiX)Ai. Nuestros resultados principales son los seguintes: Si A es positiva bajo transposición parcial (PPT) o simétrica con coeficientes positivos (SPC) o invariante bajo realineamiento luego FA ○GA :Mk →Mk es completamente reducible...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

A reduction of the separability problem to SPC states in the filter normal form

It was recently suggested that a solution to the separability problem for states that remain positive under partial transpose composed with realignment (the so-called symmetric with positive coefficients states or simply SPC states) could shed light on entanglement in general. Here we show that such a solution would solve the problem completely. Given a state in $\mathcal{M}_k\otimes\mathcal{M}_m$, we build a SPC state in $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$ with the same Schmidt number. It is known that this type of state can be put in the filter normal form retaining its type. A solution to the separability problem in $\mathcal{M}_k\otimes\mathcal{M}_m$ could be obtained by solving the same problem for SPC states in the filter normal form within $\mathcal{M}_{k+m}\otimes\mathcal{M}_{k+m}$. This SPC state can be built arbitrarily close to the projection on the symmetric subspace of $\mathbb{C}^{k+m}\otimes\mathbb{C}^{k+m}$. All the information required to understand entanglement in $\mathcal{M}_s\otimes\mathcal{M}_t$ $(s+t\leq k+m)$ lies inside an arbitrarily small ball around that projection. We also show that the Schmidt number of any state $\gamma\in\mathcal{M}_n\otimes\mathcal{M}_n$ which commutes with the flip operator and lies inside a small ball around that projection cannot exceed $\lfloor\frac{n}{2}\rfloor$.Comment: A new section was adde

Teoria de Perron Frobenius para mapas positivos

In this work we study Perron-Frobenius theory for positive maps acting on the matrix algebra $M_k$ and its subalgebras VMkV={ VXV | X pertencendo à Mk}. We show that the spectral radius of any positive map belongs to its spectrum and associated to this eigenvalue there is a positive semidefinite hermitian eigenvector. Moreover, if the map is irreducible then we show that the geometric multiplicity of the spectral radius is 1 and the image of the associated eigenvector coincides with the image of V. We also study positive maps which are direct sum of irreducible maps and we provide an indirect way to construct them.Trabalho de Conclusão de Curso (Graduação)Nesse trabalho estudamos a teoria de Perron-Frobenius para mapas positivos atuando na álgebra de matrizes Mk e nas suas sub-álgebras VMkV={ VXV | X pertencendo à Mk}. Apresentamos demonstrações dos teoremas principais dessa teoria. Mostramos que para todo mapa positivo atuando em VMkV o seu raio espectral é um autovalor e associado a ele existe um autovetor hermitiano positivo semidefinido. Além disso, se o mapa é irredutível então a multipilicidade geométrica é 1 e a imagem do autovetor associado coincide com a imagem de V. Também estudamos mapas que são soma direta de irredutíveis e mostramos uma maneira indireta de construí-los

Eventually entanglement breaking Markovian dynamics: structure and characteristic times

Funder: Cantab Capital Institute for the Mathematics of InformationWe investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT-square conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert-Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincar\'e inequalities for nonprimitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

Funder: Cantab Capital Institute for the Mathematics of InformationAbstract: We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT2 conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert–Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest