  This work is concerned with the issue of applications of the semi-definite
programming (SDP) in the field of quantum information science. Our results of
the analysis of certain quantum information protocols using this optimization
technique are presented, and an implementation of a relevant numerical tool is
introduced. The key method used is NPA discovered by Navascues et al. [Phys.
Rev. Lett. 98, 010401 (2007)].
  In chapter 1 a brief overview of mathematical methods used in this work is
presented. In chapter 2 an introduction to quantum information science is
given. Chapter 3 concerns the device-independent (DI) and
semi-device-independent (SDI) approaches in quantum cryptography. In chapter 4
our results regarding quantum information protocols which we developed using
SDP are described. The results include a new type of quantum key distribution
protocol based on Hardy's paradox; several protocols for quantum random number
expansion; a method of formulation of SDP relaxations of SDI protocols; and a
DI Santha-Vazirani source of randomness amplification protocol. In chapter 5 an
introduction to interior point methods is given. In chapter 6 the
implementation of our numerical tool dedicated to problems occurring in chapter
4 are discussed and compared with other solutions.
