Quantum Walks And Entropic Uncertainty Relations

Quantum Mechanics and Classical Information Theory are among the greatest intellectual achievements in science.Their interdisciplinary activities give birth to a promising research area:Quantum Information Theory.This thesis aims at broadening our understanding on two interesting topics of this area:discrete quantum walks and entropic uncertainty relations.It is a contribution to quantum information theory from the perspective of computer science.In the first part,we study the discrete quantum walks.Quantum walks are the quantum counterpart of random walks and are important tool for designing quantum algorithms.Aside from the algorithmic applications,quantum walks shed important insights on quantum mechanics,and for that reason alone they are worth pursuing.Our main contribution to this topic are the detailed mathematical analysis of the properties of several different kinds of one-dimensional quantum walks.These properties turn out to be helpful for designing quantum walk based quantum algorithms.In Chapter 3,we study three-state quantum walks on a line with one and two absorbing boundaries.We present analytical expressions for the absorbing probabilities using the combinatorial approach.We observe an interesting oscillating localization effect,uniquely in this model.Our theoretical results are perfectly matched with numerical simulations and complement the research of three-state quantum walks.In Chapter 4,we go further to consider the lackadaisical quantum walk model where each node has many self-loops.We derive analytic expression for the localization probability in the asymptotic limit,obtain the peak velocities of the walker,and obtain a long-time approximation for the entire probability density function.As application,we prove that lackadaisical quantum walks spread ballistically,and give an analytic solution for the variance of the walker’s probability distribution.In the second part,we study the uncertainty relation in quantum mechanics.Heisen-berg uncertainty principle states that even if we have full information about the state of a quantum system,it is impossible to predict the outcomes of incompatible measure-ments with certainty.Later,it is recognized that the side information of a quantum ob-server can help,and lead to an interplay between uncertainty and quantum correlations.Our main contribution to this topic are several novel entropic uncertainty relations for different kinds of measurements and their applications on entanglement detection.In Chapter 5,using conditional collision entropy as uncertainty measure,we investigate entropic uncertainty relations with quantum side information,for complete set of mutu-ally unbiased measurements and generalized symmetric informationally complete mea-surements.We obtain an equality relating the amount of uncertainty of measurement outcomes and the amount of entanglement of the state being measured.We outline several novel applications of our result.In Chapter 6,we develop a new kind of entan-glement detection method within the framework of marjorization uncertainty relations,on which the uncertainty is quantified by majorization.By virtue of majorization uncer-tainty bounds,we are able to construct the entanglement criteria which have advantage over the scalar detecting algorithms.