Course contents:

1. INTRODUCTION AND OVERVIEW. Information is physical. Quantum information vs. Classical information. Entanglement as a resource for quantum information. Classic and quantum Turing machines. The quantum computer. Why should theoretical physicists pay attention to quantum computers?

2. FUNDAMENTALS OF QUANTUM THEORY. States and sets (“Ensembles”). Density operators. Formalism of the density matrix. Measures and evolution. Orthogonal measures. Generalized measures. Measures of projection operators. Neumark's theorem. Superoperators.

3. THE QUANTUM BIT. Characterization of states of a qubit, Hamiltonians and measurements. Spherical representation of Bloch. Physical realizations of qubits.

4. QUANTUM ENTANGLEMENT. Composite systems. Product states vs. entangled states. Local operations and classic communication. Entanglement operations. Entanglement measures. Bipartite systems. Decomposition of Schmidt. Reduced density matrix. States of maximum entanglement. Entanglement and mixing. Entropy of entanglement.

5. THE TWO QUBITS SYSTEM. Characterization of entangled states of two qubits. Bell States. Physical realization of Bell pairs with pairs of photons or atoms.

6. USES OF THE ENTANGLEMENT. Possible machines. Dense coding. Quantum teleportation. Quantum cryptography Impossible machines Communication slower than light. The no cloning theorem.

7. BELL INEQUALITIES. Einstein locality and hidden variables. The paradox of Einstein, Podolski and Rosen. No separability of EPR pairs. Hidden quantum information. Bell inequalities. Experimental test of Bell's inequalities. The Aspect’s experiment.

8. QUANTUM COMPUTING. Classical computing vs. Quantum computing. Computational complexity. Quantum circuits. Quantum gates. Universal quantum gates. Quantum algorithms The Deutsch algorithm. The quantum database search algorithm of Grover. The Shor factorization algorithm.

9. QUANTUAM SIMULATION. The Feynman quantum simulator. The complexity of quantum simulation. Physical realization of a quantum simulator. Example: ultracold atoms in optical networks. The Hubbard tool. Quantum simulation of Bell pairs. Quantum simulation of phases of many strongly correlated bodies.

10. TOPOLOGICAL QUANTUM COMPUTING. Quantum error correction. Faulty safe quantum computing. Quantum codes. Topological order. Examples of topological phases in nature. Fractional quantum Hall effect systems. Anions. Abelian and non-Abelian anions. Kitaev’s Toric code. The topological quantum bit.

11. ENTANGLEMENT IN MANY BODY SYSTEMS. Entanglement as a useful tool to characterize quantum phases and quantum phase transitions. The area law of entanglement entropy. Renormalization group of the density matrix and tensor networks. Applications to spin lattice systems.