Mathematical Problems in Quantum Information Theory

在这个项目中，拟议研究的主要目标是开发所需的数学工具来证明量子信息理论中的几个杰出的假设。中心猜想指出最小输出熵对于乘积通道是可加的，其中通道是噪声量子系统的数学表示。数学设定涉及矩阵代数上完全正映射的性质。基于早期工作的结果和方法，PI希望为所有量子比特映射建立这种猜想，并在此过程中开发适用于更高维映射的工具。PI期望产生既有趣又对量子信息理论有用的结果。量子信息理论关注于探索物理系统中可用的新资源，这些系统的行为全部或部分由量子效应控制，例如单原子系统和单光子状态。最近的理论计算表明，这样的系统可能具有非凡的特性。一个例子是量子计算机，这是一种理论上的设备，能够超越任何标准计算机。另一个例子是无条件安全加密的协议，它可以通过将信息编码为量子态来实现。数学在这些新思想的发展中起着重要的作用。目前的建议旨在利用先进的数学技术来探索在通信系统中使用纠缠量子态的含义。一个基本问题是确定这样一个系统的信息容量，从而找到香农关于噪声信道容量的著名表达式的量子类比。拟议活动的更广泛影响在于量子信息理论在物理学和计算机科学中的潜在应用，and ultimately最终intechnology技术.

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.