Finite-Block-Length Analysis in Classical and Quantum Information Theory

This is a review article of finite-block-length analysis in classical and quantum information theory for non-specialist. Transmitting an information is a fundamental technology. However, there are several demands for this transmission. The research area to study such problems is called information theory. In the information transmission, the information is transmitted via a physical media. Hence, the analysis of this problem might depends on the property of the physical media. Indeed, while it is ideal that the analysis does not depend on this property, it depends on the following classification of the physical media. Currently, we have two kinds of physical objects, the first one is a macroscopic object, i.e., an object subject to classical physics, and the second one is a microscopic object, i.e., an object subject to quantum physics. Since these two objects have completely different behaviors, we need to build up information theory dependently of these two information media. That is, we have classical information theory and quantum information theory. In both information theory, there are very elegant theoretical results with an ideal assumption. That is, we often assume that infinitely large size of the system is available while the real situation does not satisfy this assumption. Hence, to discuss the real case, we need to care about the finite size effect of the system size. This paper reviews this finite size effect in classical and quantum information theory with respect to various topics.