Additivity Problems in Quantum Information Theory

点击翻译按钮获取中文摘要

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.