Additivity Problems in Quantum Information Theory

这个建议涉及量子信息论中的几个相关问题，所有这些问题都是由对噪声量子信道的研究引起的。第一个问题是证明单位量子比特信道的最小熵的可加性和最大迹范数的可乘性。这一结果暗示了这种信道的Holevo容量的可加性，从而解决了一个长期存在的猜想。第二个问题是研究有噪声量子信道的Shannon容量与P.I.的早期工作中引入的一个新量之间的一个严格的等式。第三个问题涉及对产物通道的一类特殊的非产物测量的研究，其目标是提供一个有趣的实验室来检验纠缠测量的想法。纳米技术的进步为科学家提供了研究和操纵微观系统的量子特性的机会，包括单原子系统和单光子态。最近的理论发现表明，这样的系统可能具有非凡的性质。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。数学在这两种思想的发展中起了关键作用。目前的建议涉及到类似的量子设备的数学研究，这些设备将用于传输和存储信息。一个基本的问题是确定这样一个系统的信息容量，从而找到香农的著名表达式的噪声信道的容量的量子类比。私家侦探描述了在一个重要的特殊情况下确定这种容量的策略，即使用“量子比特”来编码信息的无记忆通道。解决方法将涉及一些新的理论思想，这些思想可能应用于量子信息论的其他领域。

This proposal concerns several related problems in quantum informationtheory, all arising from the study of noisy quantum channels. The firstproblem is to prove additivity of minimal entropy and multiplicativity ofthe maximal trace norms for a unital qubit channel. This result would implyadditivity of the Holevo capacity for such channels, thereby settling oneparticular case of a longstanding conjecture. The second problem is toinvestigate a conjectured equality between the Shannon capacity of a noisyquantum channel and a new quantity which was introduced in earlier work ofthe P.I.. The third problem involves a study of one special class ofnon-product measurements for product channels, with the goal of providingan interesting laboratory for testing ideas about entangled measurements.Advances in nanotechnology have provided the opportunity for scientists tostudy and manipulate the quantum properties of microscopic systems,including single-atom systems and single-photon states. Recent theoreticaldiscoveries indicate 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 played a key role in thedevelopment of both of these ideas. The current proposal concerns a similarmathematical investigation of quantum devices which would be used totransmit and store information. 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. TheP.I. describes a strategy for determining this capacity in an importantspecial case, namely a memoryless channel where "qubits" are used to encodethe information. The method of solution would involve some new mathematicalideas which may have applications to other areas of quantum informationtheory.