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This work is concerned with the issue of applications of the semi-definiteprogramming (SDP) in the field of quantum information science. Our results ofthe analysis of certain quantum information protocols using this optimizationtechnique are presented, and an implementation of a relevant numerical tool isintroduced. 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 ispresented. In chapter 2 an introduction to quantum information science isgiven. Chapter 3 concerns the device-independent (DI) andsemi-device-independent (SDI) approaches in quantum cryptography. In chapter 4our results regarding quantum information protocols which we developed usingSDP are described. The results include a new type of quantum key distributionprotocol based on Hardy's paradox; several protocols for quantum random numberexpansion; a method of formulation of SDP relaxations of SDI protocols; and aDI Santha-Vazirani source of randomness amplification protocol. In chapter 5 anintroduction to interior point methods is given. In chapter 6 theimplementation of our numerical tool dedicated to problems occurring in chapter4 are discussed and compared with other solutions.

Applications of semi-definite optimization in quantum information protocols