Nonlocality & Communication Complexity

This thesis discusses the connection between the nonlocal behavior of quantum mechanics and the communication complexity of distributed computations. The first three chapters provide an introduction to quantum information theory with an emphasis on the description of entangled systems. The next chapter looks at how to measure the complexity of distributed computations. This is expressed by the ‘communication complexity’, defined as the minimum amount of communication required for the evaluation of a function —a communication necessary because the input strings and are distributed over separated parties. In the theory of quantum communication, we try to use the nonlocal effects of entangled quantum bits to reduce communication complexity. In chapters 5, 6 and 7, such an improvement over classical communication is indeed established for various functions. However, it is also shown that entanglement does not lead to a more efficient calculation of the inner product function. We thus reach the conclusion that nonlocality sometimes—but not always—allows a reduction in communication complexity. This subtle relationship between nonlocality and communication vanishes when we consider ‘superstrong’ correlations. We demonstrate that if a violation of the Clauser-Horne-Shimony-Holt inequality with the maximum factor of is assumed, all decision problems have the same trivial complexity of a single bit. The thesis concludes with an overview of the current status of quantum communication theory, and a discussion of the experimental feasability of the suggested protocols.