Information Theory and Quantum Information Theory

- Information Theory

Information theory is a mathematical approach that studies how information is processed and measured during transmission. It's also known as the mathematical theory of communication. 

Information theory is concerned with the quantification, storage, and communication of information. It's used to analyze communication systems and has applications in linguistics, physiology, and physics. 

The field of information theory was established in the 1920s by Harry Nyquist and Ralph Hartley, and in the 1940s by Claude Shannon. Shannon is known as the "father of information theory". 

Information theory deals with the extent of information related to random variables. The main quantities of information are mutual information and entropy. Mutual information measures the information in common between two variables, while entropy measures the information in one random variable.

Please refer to Wikipedia: Information Theory for more details.

- Quantum Information Theory

Quantum information theory (QIT) is a generalization of classical information theory to use quantum-mechanical particles and interference. It is used in the study of quantum computation and quantum cryptography. 

QIT is a field of study that combines information theory and quantum mechanics to study the transmission, analysis, and processing of information. It's a subject of intense research that lies at the intersection of physics, computer science, and mathematics. 

QIT studies the meaning and limits of communicating classical and quantum information over quantum channels. It examines the information-processing capabilities of quantum systems. For example, a pair of quantum systems in an entangled state can be used as a quantum information channel to perform tasks that are impossible for classical systems. 

QIT is similar to its classical counterpart, classical information theory, which is the mathematical theory of information acquisition, storage, transmission, and processing. 

One example of quantum information is factoring, or finding the prime numbers that multiply together to make another number. While there is only one way to factor any number, factoring large numbers is a very hard problem on classical computers. On a quantum computer, it's relatively easy.

Please refer to Wikipedia: Quantum Information for more details.

- Discrete Mathematics

Discrete mathematics is a branch of mathematics that studies countable or distinct mathematical structures. Examples of discrete structures include graphs, combinations, and logical statements. Discrete structures can be finite or infinite. 

Discrete mathematics excludes topics in "continuous mathematics" such as calculus, real numbers, or Euclidean geometry. Discrete objects can often be characterized by integers, while continuous objects require real numbers. 

Some examples of discrete mathematics include: 

• Graph theory: The study of mathematical structures used to model pairwise relations between objects, such as social connections or networks. A graph is a mathematical structure that represents a particular function by connecting a set of points.
  
• Combinatorics: The study of how discrete objects combine with one another and the probabilities of various outcomes. For example, one may ask, in how many ways can we form a five-letter word.
  
• Algorithms: Examples of common discrete mathematics algorithms include searching algorithms, sorting algorithms, and insertion and deletion algorithms. 

Please refer to Wikipedia: Discrete Mathematics for more details.