Additivity Problems in Quantum Information Theory

这一建议涉及量子信息论中的几个相关问题，这些问题都源于对噪声量子信道的研究。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。这一结果将意味着这种信道的Holevo容量是可加性的，从而解决了一个长期存在的猜想的特殊情况。第二个问题是研究有噪声量子信道的香农容量与P.I.早期工作中引入的一个新量之间的猜想相等。第三个问题涉及对一类特殊的产品渠道的非产品测量的研究，目的是提供一个有趣的实验室来测试关于纠缠测量的想法。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性质的机会，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。量子计算机就是一个例子，它是一种能够超越任何标准计算机的理论设备。另一个例子是无条件安全加密协议，这将通过将消息编码为量子态来实现。数学在这两种思想的发展中发挥了关键作用。目前的提议涉及对用于传输和存储信息的量子设备进行类似的数学研究。一个基本的问题是确定这样一个系统的信息容量，从而找到香农著名的噪声信道容量表达式的量子类比。P.I.描述了一种在重要的特殊情况下确定这种容量的策略，即使用“量子比特”对信息进行编码的无记忆信道。这种求解方法将涉及一些新的数学思想，这些思想可能会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.