## The Physics of Classical and Quantum Information

*Objective*: This course aims at offering graduate students an introduction to the field of the Physics of Information, namely to the emergent area of Quantum Information Theory, covering both its physical foundations and its revolutionary applications to computation, cryptography and telecommunications, and their respective technologies.

*Target*: Students of the doctoral programmes in Physics and Technological Physics Engineering. Also relevant for doctoral students in computer science, information sciences, electrical engineering and mathematics.

*Format*: Lectures including problem classes (7,5 credits ECTS).

*Lecturer*: Yasser Omar

*Syllabus*:

1. The Physics of Information

1.1. Landauer’s Principle and the thermodynamics of classical information

1.2. The current limits of classical information processing

1.3. Preview of the main challenges and results presented in this course

2. Quantum Mechanics: essential results and novelties

2.1. Particles and states

2.2. Postulates and mathematical formalism

2.3. Example: the quantum bit

2.3.1. Preparation, evolution, observation

2.3.2. Bloch representation

2.3.3. Experimental implementations

2.4. Mixed states and density operator

2.5. Generalized measurements and discrimination of quantum states

3. Quantum Entanglement

3.1. The Einstein-Podolsky-Rosen problem and Bell inequalities

3.2. Other inequalities, GHZ states and experimental tests

3.3. Bell states and quantum correlations

3.4. Examples: teleportation and dense coding of information

3.5. Definition and classification of entangled states: Schmidt decomposition, von Neumann entropy

3.6. Entanglement concentration and purification

3.7. Applications of entanglement

4. Quantum Information Theory

4.1. Main results of Classical Information Theory

4.2. Shannon entropy, von Neumann entropy and other measures of information

4.3. Schumacher’s theorem

4.4. Accessible information and Holevo bound

4.5. Noise and loss of coherence

4.6. No-cloning theorem

4.7. Quantum communications and respective implementations

5. Quantum Computation

5.1. Elementary concepts of classical computation

5.2. Logic gates and quantum circuits

5.3. Deutsch’s problem

5.4. Quantum algorithms: Shor, Grover and others

5.5. Quantum complexity

5.6. Errors and correcting codes

5.7. Possible implementations for the quantum computer

6. Quantum cryptography

6.1. Basic concepts of classical cryptography

6.2. Quantum key distribution:

6.2.1. With single particles – BB84 and BB92 protocols

6.2.2. With entangled particles – Ekert91 protocol

6.3. Eavesdropping and security in communications

6.4. Implementations with photons and commercial applications

7. Conclusion

7.1. Revision of the results presented in the course: a global perspective

7.2. Extensions of theses results and what was left to teach

7.3. Current challenges in the field of Quantum Information