Maximally entangled states from symmetric and alternating groups

Abstract

Uno de los propósitos fundamentales de la teoría de la información tiene que ver con la construcción de protocolos que permitan intercambiar datos entre dos o más partes. Sin embargo, estos protocolos están sujetos a la interacción con el medio ambiente, lo cual los hace propensos a errores. Debido a las propiedades de enredamiento de los sistemas cuánticos de N partículas, los sistemas localmente máximamente enredados permiten recuperar el estado de un sistema si ocurre un error en alguna de sus N partes al pasarlo por cierto canal cuántico. A pesar de ello, esos estados no existe una forma canónica de construir dichos estados. Teniendo en cuenta la importancia de los estados localmente máximamente enredados, en esta tesis de maestría propongo un método que permite etiquetar estos estados a partir de productos tensoriales de representaciones de grupos finitos, y en particular del grupo simétrico. A partir de éstas representaciones, además, es posible clasificar estados compuestos por N copias del mismo sistema cuántico de d niveles de acuerdo a sus propiedades de simetrización, dadas por la dualidad de Schur-Weyl

One of the main goals pursued by Quantum Information Theory is to provide secure protocols to share data between different parties, such that the information sent cannot be easily intercepted by an eavesdropper. Several of these protocols limit the information that can be obtained by one of the parties when they try to recover the original message individually, such as Quantum Secret Sharing schemes (QSS). In these schemes, any individual party or any selected subgroup of parties is incapable of recovering any portion of the original quantum state sent. Since locally maximally entangled states (LME states) serve these purposes, the search for this kind of quantum states have become one of the most explored areas in this field. There is no canonical method to find the set of all possible maximally entangled states of N parties. In fact, finding SLOCC (stochastic local operations and classic comunication) orbits of entanglement classes is still an open task for N 4. Fortunately, states of this type can be constructed by projecting any multipartite state over invariant one-dimensional subpaces of tensor products of irreducible representations of finite groups, such as the symmetric group Sn. Moreover, these can be found to have some symmetry properties given by how they transform under the action of the symmetric group SPN acting over its N parties. In this sense, the current thesis proposes a more intuitive form of conceiving such states. It provides a form of constructing maximally entangled states from tensor products of representations of the symmetric group Sn and the alternating group An. Each of these states can be labeled in terms of the irreducible action of the symmetric group SN over the N indices of a multipartite state. In fact, this thesis provides a form of reducing the dimensionality of the problem when self-adjoint representations of Sn are used, given some relations between this group and the representations of the alternating group