In this project, the principal investigator will pursue two major themes: (1) the Kadison-Singer problem and (2) the theory of tensor products of operator systems. The Kadison-Singer problem is a major problem in this area of mathematics that has been unsolved since 1954. The principal investigator has made some recent progress on the problem and will explore three new avenues of attack on it. Operator systems and completely positive maps play a central role in several areas of mathematics, including quantum computing, quantum information theory, and applications to C*-algebras and von Neumann algebras. The principal investigator will continue to develop the general tensor theory of operator systems and continue to apply this theory to problems in quantum information and quantum computing.
The area of mathematics known as frame theory is concerned with systems that are used to sample signals of various types and then to reconstruct the signals from the samples, such as one does when sampling a soundwave, burning it to a CD, then playing back the music from the CD. Engineers always build redundancy (or oversampling) into such systems in order to ameliorate the effects of errors in the numerical values of the samples. Progress on the Kadison-Singer problem should translate to a more precise understanding of how redundancy behaves than exists at the present time. Roughly, it asks whether or not systems with "finite redundancy" can always be divided into finitely many systems with no redundancy. The second goal of the project is concerned with developing the mathematics of quantum information theory and quantum computing. Although no one can predict whether or not quantum computers will ever be built, if they are, it is certainly vital to the national interests to have developed sufficient human resources to be competitive in putting them to use. Consequently, the principal investigator's students are introduced to this area, and his work on the tensor theory of operator systems has applications to questions about parallel structure in the quantum setting.
