DOI: https://doi.org/10.15446/recolma.v50n2.62210

Quantum Information and the Representation Theory of the Symmetric Group

Información Cuántica y la Teoría de Representación del Grupo Simétrico

Alonso Botero

Universidad de los Andes, Bogotá, Colombia. abotero@uniandes.edu.co

Abstract

A number of important results in quantum information theory can be connected quite elegantly to the representation theory of the symmetric group through a quantum analogue of the classical information-theoretic "method of types" that arises naturally from the Schur-Weyl duality. We will give a brief introduction to this connection and briefly discuss some of the results that follow from it, such as quantum source compression rates, entanglement concentration rates, quantum entropy inequalities, and the admissisble spectra of partial density matrices from pure, multipartite entangled states.

Keywords: Representation Theory, Quantum Information Theory, Schur-Weyl duality, Quantum Shannon theorem, Entanglement concentration.

Resumen

Un gran número de resultados importantes en la teoría de la información cuántica se pueden conectar con la teoría de la representación del grupo simétrico, a través de un análogo cuántico del llamado método de tipos que emerge de manera natural de la dualidad de Schur-Weyl. En este artículo daremos una breve introducción a esta conexión y discutiremos algunos resultados que emergen de la misma, como son las tasas de compresión de a fuente cuántica, tasas de concentración de enredamiento, desigualdadades de la entropía cuántica, y condiciones sobre los espectros admisibles de matrices parciales de densidad provenientes de un estado cuántico puro multipartito.

Palabras claves: Teoría de la representación, Teoría de la información cuántica, dualidad de Schur-Weyl, Teorema de Shannon cuántico, Concentración de enredamiento.

Mathematics Subject Classification: 20G05, 94A15, 81P45, 60F10.

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Recibido: julio de 2016 Aceptado: noviembre de 2016