The primary goal of the proposed research in this project is to develop the mathematical tools needed to prove several outstanding conjectures in Quantum Information Theory. The central conjecture states that minimal output entropy is additive for product channels, where a channel is the mathematical representation of a noisy quantum system. The mathematical setting concerns properties of completely positive maps on matrix algebras. Building on results and methods from earlier work, the PI expects to establish this conjecture for all qubit maps, and in the process develop tools that will apply to higher dimensional maps. The PI expects to produce results that are both mathematically interesting and also useful for quantum information theory.
Quantum Information Theory is concerned with exploring the new resources that are available in physical systems whose behavior is wholly or partly governed by quantum effects, for example single-atom systems and single-photon states. Recent theoretical discoveries indicate that such systems may have extraordinary properties. One example is the quantum computer, which is a theoretical device capable of outperforming any standard computer. Another example is a protocol for unconditionally secure encryption, which would be achieved by encoding messages as quantum states. Mathematics has played an essential role in the development of these new ideas. The present proposal is aimed at using advanced mathematical techniques to explore the implications of using entangled quantum states in communication systems. A fundamental problem is to determine the information capacity of such a system, and thereby find the quantum analog of Shannon's famous expression for the capacity of a noisy channel. The broader impact of the proposed activity rests on the potential applications of quantum information theory in physics and computer science, and ultimately in technology.
