Aspects Of Quantum Channel Capacities

Channel capacity, which characterizes the communication channel's capability tofaithfully transmit information against noises, is the most fundamental theoretical prob-lem in the subject of communication. Claude Shannon, a mathematician who estab-lished the Information Theory, provided a beautifully simple formula for the capacityof a classical channel, widely known as Shannon's Second Law. Our world howeveris at a basic level described by quantum mechanics. To understand the ultimate limitthe laws of physics impose on our ability to communicate information, the underlyingquantum behavior of the channels should be considered. A quantum channel can notonly convey classical messages, but also quantum data, it can also carry classical privateinformation. Naturally, deriving capacity formulae(namely, classical capacity, privatecapacity and quantum capacity) for quantum channels is a central task of quantum in-formation theory. Despite considerable progresses, tractable formulae for the classical,private and quantum capacities are still out of reach. The work of this thesis, as aneffort and achievement on the route of understanding the problem of quantum channelcapacities, focuses on the following two issues.1. In 2008, Smith from IBM Rearch, and Yard from LANL, have found that thequantum capacity of quantum channels is not generally additive. Specifically, two dif-ferent channels, when combined together, can transmit quantum information more thanthe sum of their individual quantum capacities. In this thesis, we further prove thatanother capacity quantity, namely the private capacity, is also not additive. This strangecounter-intuitive feature is impossible for Shannon's classical channel capacity. Ourfinds demonstrate that, the ability of a quantum channel to transmit information doesnot only depends on the channel itself, but also has much to do with the context wherethe channel is used. The capacities, however, as measures of the information transmit-ting capability, is not so intrinsic in the quantum world as that of Shannon's classicalchannels. In this work, we also yield another counterexample to the additivity of quan-tum capacity, of which the underlying reasoning is different from that of Smith andYard's. Because, unlike theirs, the nonadditivity in our construction does not comefrom the individual channel's ability to transmit private information.2. Network information theory, which deals with the capacity problems of multi-user channels, is an important component in classical information theory. In quantum information theory, problems of information transmitting capacity of multi-user chan-nels have also attracted considerable attentions. Together with Dr. Fre′de′ric Dupuisand Prof. Patrick Hayden, we investigate the capacities of quantum broadcast chan-nels. Without loss of generality, we only deal with the case of two receivers. Firstly,we think about independent quantum information transmission from the sender to thetwo receivers, assisted by free entanglement. We obtain a single-letter expression forthe achievable rate region, whose regularization form is confirmed to be the capacityregion by proving the converse theorem. Secondly, we derive a capacity region ex-pression for independent quantum information transmission without assistance of freeentanglement, and for independent classical information transmission with assistanceof free entanglement, based on the first achievement of the entanglement-assisted quan-tum capacity region. Lastly, we present a single-letter example: the so called classicaldeterministic channels. We find that, for such channels, the two entanglement-assistedcapacity regions can be expressed by single-letter formulae, and regularization is notneeded.