Quantum information theory

Schumann, Robert Helmut (2000-12)

Thesis (MSc)--Stellenbosch University, 2000

Thesis

ENGLISH ABSTRACT: What are the information processing capabilities of physical systems? As recently as the first half of the 20th century this question did not even have a definite meaning. What is information, and how would one process it? It took the development of theories of computing (in the 1930s) and information (late in the 1940s) for us to formulate mathematically what it means to compute or communicate. Yet these theories were abstract, based on axiomatic mathematics: what did physical systems have to do with these axioms? Rolf Landauer had the essential insight - "Information is physical" - that information is always encoded in the state of a physical system, whose dynamics on a microscopic level are well-described by quantum physics. This means that we cannot discuss information without discussing how it is represented, and how nature dictates it should behave. Wigner considered the situation from another perspective when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences". Why are the computational techniques of mathematics so astonishingly useful in describing the physical world [1]? One might begin to suspect foul play in the universe's operating principles. Interesting insights into the physics of information accumulated through the 1970s and 1980s - most sensationally in the proposal for a "quantum computer". If we were to mark a particular year in which an explosion of interest took place in information physics, that year would have to be 1994, when Shor showed that a problem of practical interest (factorisation of integers) could be solved easily on a quantum computer. But the applications of information in physics - and vice versa - have been far more widespread than this popular discovery. These applications range from improved experimental technology, more sophisticated measurement techniques, methods for characterising the quantum/classical boundary, tools for quantum chaos, and deeper insight into quantum theory and nature. In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems. In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems. The final chapter presents a new resource: quantum information. This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe.

AFRIKAANSE OPSOMMING: Tot watter mate kan fisiese sisteme informasie verwerk? So onlangs soos die begin van die 20ste eeu was dié vraag nog betekenisloos. Wat is informasie, en wat bedoel ons as ons dit wil verwerk? Dit was eers met die ontwikkeling van die teorieë van berekening (in die 1930's) en informasie (in die laat 1940's) dat die tegnologie beskikbaar geword het wat ons toelaat om wiskundig te formuleer wat dit beteken om te bereken of te kommunikeer. Hierdie teorieë was egter abstrak en op aksiomatiese wiskunde gegrond - mens sou wel kon wonder wat fisiese sisteme met hierdie aksiomas te make het. Dit was Rolf Landauer wat uiteindelik die nodige insig verskaf het - "Informasie is fisies" - informasie word juis altyd in 'n fisiese toestand gekodeer, en so 'n fisiese toestand word op die mikroskopiese vlak akkuraat deur kwantumfisika beskryf. Dit beteken dat ons nie informasie kan bespreek sonder om ook na die fisiese voorstelling te verwys nie, of sonder om in ag te neem nie dat die natuur die gedrag van informasie voorskryf. Hierdie situasie is vanaf 'n ander perspektief ook deur Wigner beskou toe hy geskryf het oor "die onredelike doeltreffendheid van wiskunde in die natuurwetenskappe". Waarom slaag wiskundige strukture en tegnieke van wiskunde so uitstekend daarin om die fisiese wêreld te beskryf [1]? Dit laat 'n mens wonder of die beginsels waarvolgens die heelal inmekaar steek spesiaal so saamgeflans is om ons 'n rat voor die oë te draai. Die fisika van informasie het in die 1970's en 1980's heelwat interessante insigte opgelewer, waarvan die mees opspraakwekkende sekerlik die gedagte van 'n kwantumrekenaar is. As ons één jaar wil uitsonder as die begin van informasiefisika, is dit die jaar 1994 toe Shor ontdek het dat 'n belangrike probleem van algemene belang (die faktorisering van groot heelgetalle) moontlik gemaak word deur 'n kwantumrekenaar. Die toepassings van informasie in fisika, en andersom, strek egter veel wyer as hierdie sleutel toepassing. Ander toepassings strek van verbeterde eksperimentele metodes, deur gesofistikeerde meetmetodes, metodes vir die ondersoek en beskrywing van kwantumchaos tot by dieper insig in die samehang van kwantumteorie en die natuur. In hierdie tesis bied ek 'n kort oorsig oor die belangrikste idees van kwantuminformasie teorie. Die eerste hoofstuk bestaan uit inleidende materiaal oor die belangrikste idees van waarskynlikheidsteorie en klassieke informasie teorie. Kwantummeganika word op 'n gevorderde voorgraadse vlak ingevoer, saam met die nodige gereedskap van kwantummeganika vir oop stelsels. In die tweede hoofstuk spreek ek die voorstelling van klassieke informasie en kwantumstelsels aan, en die gepaardgaande moontlikhede vir 'n agent wat informasie uit sulke stelsels wil kry. Die laaste hoofstuk ontgin 'n nuwe hulpbron: kwantuminformasie. Gedurende die afgelope tien jaar het hierdie nuwe hulpbron tot verbysterende nuwe toepassings gelei en ons keer op keer tot onverwagte nuwe insigte oor kwantumteorie en die heelal gelei.

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