Classical and Nonextensive Information Theory

Information theory deals with measurement and transmission of information through a channel. A fundamental work in this area is the Shannon’s Information Theory (see [2], Chapter 11), which provides many useful tools that are based on measuring information in terms of the complexity of structures needed to encode a given piece of information. Shannon’s theory solves two central problems for classical information: (1) How much can a message be compressed; i.e., how redundant is the information? (The noiseless coding theorem). (2) At what rate can we communicate reliably over a noisy channel; i.e., how much redundancy must be incorporated into a message to protect against errors? (The noisy channel coding theorem). In this theory, the information and the transmission channel are formulated in a probabilistic point of view. In particular, it established firmly that the concept of information has to be accepted as a fundamental, logically sound concept, amenable to scientific scrutiny; it cannot be viewed as a solely anthropomorphic concept, useful in science only on a metaphoric level. If the channel is a quantum one or if the information to be sent has been stored in quantum states than Quantum Information Theory starts. The fact that real systems suffer from unwanted interactions with the outside world makes the systems undergo some kind of noise. It is necessary to understand and control such noise processes in order to build useful quantum information processing systems. Quantum Information Theory basically deals with tree main topics: 1.Transmission of classical information over quantum channels 2.Quantifying quantum entanglement for quantum information 3.Transmission of quantum information over quantum channels. The aim of this work is to extend The Noiseless Channel Coding Theorem for a nonextensive entropic form due to Tsallis [4]. Further works in Quantum Information may be also provided as a consequence of this research. To achieve this goal, we review the development of Shannon’s Theory presented in [2], pp. 537. In this reference, a central definition is an ǫ− typical sequence. Some results in the Shannon’s Theory can be formulated by using elegant properties about these sequences. Thus, after reviewing some results about ǫ − typical sequences, we try to generalize these results by using Tsallis entropy, a kind of nonextensive entropy. That is the key idea of this work. Some results about Shannon’s Theory generalizations, by using Tsallis entropy, can be also found in [5]. However, that reference follows a different approach. In section 2 we review the classical information theory. Then, in section 3, we present a preliminary result by using nonextensive theories. The Central Limit Theorem, used during the presentation that follows, is developed on section 5 in order to complete the material.