Abstract

Optimal rates of quantum information-processing tasks, such as compression and transmission of information, and manipulation of entanglement, were initially obtained in the so-called “asymptotic, memoryless setting”. In this setting, the underlying resources (e.g. information sources, channels and entanglement resources) are assumed to be memoryless, and to be available for asymptotically many uses. The optimal rates were shown to be given in terms of entropic quantities expressible in terms of the quantum relative entropy.

In real-world communications and cryptographic systems, this setting is not generally valid: resources are used a finite number of times, and there may be correlations between their successive uses. Hence, it is important to evaluate the fundamental limits on information-processing tasks in the “one-shot setting”, in which one considers a finite number of uses of arbitrary resources.

In the first lecture, I will discuss entropic quantities which play an important role in the one-shot setting, highlighting their salient mathematical properties and operational significances.

In the second lecture, I will focus on the problem of transmission of information through quantum channels, to illustrate how some of the entropic quantities (introduced in the first lecture) arise naturally in the one-shot setting.