Universal Information Processing

Abstract

In a practical setting, it is not easy to perfectly identify the channel or the information source. To avoid this problem in classical information theory, we can employ universal code that works independently of the channel or the information source. In particular, Csiszár and Körner established the method of types for universal code, where the universality is the independence of the protocol from the channel or the information source. Schur duality is the joint representation of the special unitary group and the permutation group and can be regarded as quantum analogue of the method of types. Firstly, this chapter explains the method of types and Schur duality to clarify how Schur duality works as a quantum analogue of the method of types. Then, we proceed to universal codes or protocols in individual topics, estimation of density matrix, hypothesis testing of quantum state, entanglement concentration, quantum data compression, and classical-quantum channel, etc. When we do not care about the universality, we can discuss these topics without use of representation theory. However, to construct protocols to achieve the universality, we need to employ Schur duality theory because they cannot be constructed without use of Schur duality.