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## Quantum Information Theory Tutorial

What are the ultimate limits that nature imposes on communication and what are effective procedures for achieving these limits? In order to answer these questions convincingly, we must reassess the theory of information under a “quantum lens.” That is, since quantum mechanics represents our best understanding of microscopic physical phenomena and since information is ultimately encoded into a physical system of some form, it is necessary for us to revise the laws of information established many years ago by Shannon. This is not merely an academic exercise, but instead represents one of the most exciting new frontiers for physics, mathematics, computer science, and engineering. Entanglement, superposition, and interference are all aspects of quantum theory that were once regarded as strange and in some cases, nuisances. However, nowadays, we understand these phenomena to be features that are the enabling fuel for a new quantum theory of information, in which seemingly magical possibilities such as teleportation are becoming reality. Two other notable examples are increased communication capacities of noisy communication channels and secure encryption based on physical principles. Concepts developed in the context of quantum information theory are now influencing other areas of physics as well, such as quantum gravity, condensed matter, and thermodynamics. Furthermore, quantum information theory has given us a greater understanding of the foundations of quantum mechanics and might eventually lead to a simpler set of postulates for quantum mechanics.

This tutorial will review the basics of quantum information, in an effort to enable those trained in the traditional formulation of information theory to have a grasp for what distinguishes quantum information theory from the traditional formulation. An outline is as follows:

- background on quantum information and connections to classical information, including density operators, channels, measurements, purification, isometric extension, coherent measurement;
- noiseless protocols of entanglement distribution, teleportation, superdense coding;
- distance measures including trace distance and fidelity. Uhlmann’s theorem and gentle measurement;
- quantum entropy and entropy inequalities;
- protocols including Schumacher compression, classical communication, entanglement-assisted classical communication, quantum communication. Nonadditivity of capacities and superactivation.

*Last modified: July 10, 2016.*