Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations

In the first part of this thesis, we discuss the algebraic approach to classical and quantum physics and develop information theoretic concepts within this setup. In the second part, we discuss the uncertainty principle in quantum mechanics. The principle states that even if we have full classical information about the state of a quantum system, it is impossible to deterministically predict the outcomes of all possible measurements. In comparison, the perspective of a quantum observer allows to have quantum information about the state of a quantum system. This then leads to an interplay between uncertainty and quantum correlations. We provide an information theoretic analysis by discussing entropic uncertainty relations with quantum side information. In the third part, we discuss the concept of randomness extractors. Classical and quantum randomness are an essential resource in information theory, cryptography, and computation. However, most sources of randomness exhibit only weak forms of unpredictability, and the goal of randomness extraction is to convert such weak randomness into (almost) perfect randomness. We discuss various constructions for classical and quantum randomness extractors, and we examine especially the performance of these constructions relative to an observer with quantum side information. In the fourth part, we discuss channel simulations. Shannon's noisy channel theorem can be understood as the use of a noisy channel to simulate a noiseless one. Channel simulations as we want to consider them here are about the reverse problem: simulating noisy channels from noiseless ones. Starting from the purely classical case (the classical reverse Shannon theorem), we develop various kinds of quantum channel simulation results. We achieve this by using classical and quantum randomness extractors that also work with respect to quantum side information.
Comment: PhD thesis, ETH Zurich. 214 pages, 13 figures, 1 table. Chapter 2 is based on arXiv:1107.5460 and arXiv:1308.4527 . Section 3.1 is based on arXiv:1302.5902 and Section 3.2 is a preliminary version of arXiv:1308.4527 (you better read arXiv:1308.4527). Chapter 4 is (partly) based on arXiv:1012.6044 and arXiv:1111.2026 . Chapter 5 is based on arXiv:0912.3805, arXiv:1108.5357 and arXiv:1301.1594