The primary focus of this project is the study of a variety of mathematical problems in quantum computation and information theory. Although building quantum computers remains a formidable experimental challenge, they have the potential to efficiently handle problems whose computational complexity puts them beyond the scope of classical digital computers. Some aspects of this proposal, such as error correction and the exploration of multi-bit gates, are directly related to quantum computation. However, the main emphasis is on problems in the related area of quantum communication, particularly channel capacity, optimal encoding and decoding schemes and classification of entanglement. The P.I.'s past work on the analysis of multi-particle systems in strong magnetic fields, the properties of quantum-mechanical entropy, and maps on operator algebras provides a strong starting point for developing new mathematical tools needed to deal with these challenging issues.
The possibility of using quantum computers to efficiently factoring large numbers, is a potential threat to the security of existing cryptographic protocols. Fortunately, using quantum particles for communication, which is more feasible experimentally, offers new mechanisms for distributing cryptographic keys as well as encoding messages and transmitting information. These include both procedures based on the transmission of quantum particles for encoding and sending information; and innovative new protocols, based on a quantum phenomenon known as entanglement, involving particles at distant sites augmented by some classical communication. As with classical communication, one must be prepared to deal with noisy channels and much of this proposal deals with the mathematics of noisy channels. Some of this work has direct implications for experimental design, since it is important to know how to best allocate resources and choose coding schemes to minimize the effects of noise.
