The Classical Capacity of a Quantum Channel

Quantum information theory studies the laws of transmission, transformation and storage of information in the systems obeying the rules of quantum physics. One of its major achievements is the creation and thorough investigation of the concept of quantum communication channels. This has resulted in an elaborated structural theory and was accompanied by the discovery of a whole spectrum of entropic quantities characterizing the information-processing performance of the channels.
The topic of this lecture – the capacity of a quantum channel for transmitting classical information – is intended to make a bridge between the classical and the quantum theories and is especially convenient for a smooth transition from the former to the latter. Moreover, being the earliest and perhaps the most mature part of quantum Shannon theory, this topic continues to develop actively. Several recent achievements mentioned in the lecture, as well as intriguing open questions, are pertinent to this line of research.
Basing on simple matrix analysis, we begin with the demonstration of a close parallelism between classical and quantum statistical descriptions of information transmission processes; on the other hand, we stress the fundamental peculiarities of the quantum description, namely “complementarity” and “entanglement” which are absent in the classical picture.
Then we introduce a basic notion of a classical-quantum channel as a channel with classical input and quantum output, and give a brief survey of a variety of the relevant results: from the analog of Shannon’s channel coding theorem to the most recent achievements concerning error exponents, higher order asymptotics and strong converses. Next, we discuss the general concept of a (quantum) channel, its algebraic structure and the classical capacities, and touch upon the remarkable quantum phenomenon of superadditivity of information in memoryless channels due to entanglement in the decoding and encoding procedures. We then describe quantum Gaussian channels and report on the progress concerning the noncommutative analogs of Shannon’s famous capacity formula based on the recent solution of the long-standing “Gaussian optimizer conjecture”.
Finally, we comment on the “zoo” of different capacities of a quantum channel. Remarkably, in the quantum case the notion of channel capacity splits, giving a whole spectrum of information-processing characteristics depending on the kind of the data transmitted (classical or quantum) as well as on the available additional resources such as entanglement assistance or the feedback.