Mathematical Problems in Quantum Information Theory

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The primary goal of the proposed research in this project is todevelop the mathematical tools needed to prove several outstanding conjecturesin Quantum Information Theory. The central conjecture states thatminimal output entropy is additive for product channels, where a channelis the mathematical representation of a noisy quantum system. Themathematical setting concerns properties ofcompletely positive maps on matrix algebras. Building on results andmethods fromearlier work, the PI expects to establish this conjecture for allqubit maps, and in the process develop tools that will apply to higherdimensionalmaps. The PI expects to produce results that are both mathematicallyinterestingand also useful for quantum information theory.Quantum Information Theory is concerned with exploring the new resources thatare available in physical systems whose behavior is wholly or partlygoverned byquantum effects, for example single-atom systems and single-photon states.Recent theoreticaldiscoveriesindicate that such systems may have extraordinary properties. One exampleis the quantum computer, which is a theoretical device capable ofoutperforming any standard computer. Another example is a protocol forunconditionally secure encryption, which would be achieved by encodingmessages as quantum states. Mathematics has played an essential role in thedevelopment of these new ideas. The present proposal is aimed at usingadvanced mathematical techniques to explore the implications ofusing entangled quantum states in communication systems.A fundamental problem is to determine theinformation capacity of such a system, and thereby find the quantum analogof Shannon's famous expression for the capacity of a noisy channel.The broader impact of the proposed activity rests on the potentialapplications ofquantum information theory in physics and computer science, and ultimately intechnology.