| Summary: | Keywords: entanglement, complexity, entropy, measurement In chapter 1 the basic principles of communication complexity are introduced. Two-party communication is described explicitly, and multi-party communication complexity is described in terms of the two-party communication complexity model. The relation to entropy is described for the classical communication model. Important concepts from quantum mechanics are introduced. More advanced concepts, for example the generalized measurement, are then presented in detail. In chapter 2 the di erent measures of entanglement are described in detail, and concrete examples are provided. Measures for both pure states and mixed states are described in detail. Some results for the Schmidt decomposition are derived for applications in communication complexity. The Schmidt decomposition is fundamental in quantum communication and computation, and thus is presented in considerable detail. Important concepts such as positive maps and entanglement witnesses are discussed with examples. Finally, in chapter 3, the communication complexity model for quantum communication is described. A number of examples are presented to illustrate the advantages of quantum communication in the communication complexity scenario. This includes communication by teleportation, and dense coding using entanglement. A few problems, such as the Deutsch-Jozsa problem, are worked out in detail to illustrate the advantages of quantum communication. The communication complexity of sampling establishes some relationships between communication complexity, the Schmidt rank and entropy. The last topic is coherent communication complexity, which places communication complexity completely in the domain of quantum computation. An important lower bound for the coherent communication complexity in terms of the Schmidt rank is dervived. This result is the quantum analogue to the log rank lower bound in classical communication complexity. === Prof. W.H. Steeb
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