Elements of quantum information theory: from entanglement to open quantum systems dynamics, via the quantum measurement process.
Theory of open quantum systems, i.e. interacting with the external environment: from quantum maps to the analysis of entropy and information, from error correction to optimal control theory, towards the new millennium quantum technologies.
- M.A. Nielsen and I.A. Chuang, "Quantum computation and quantum information", Cambridge University Press (2003).
- M.W. Wilde, "Quantum Information Theory", Cambridge University Press (2013).
- H.-P. Breuer and F. Petruccione, "The theory of open quantum systems", Oxford University Press (2002).
- I. Bengtsson and K. Zyczkowski, "Geometry of quantum states", Cambridge University Press (2006).
- P. Kaye, R. Laflamme, M. Mosca, "An introduction to Quantum Computing", Oxford University Press (2007).
- T. Heinosaari and M. Ziman, "The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement", Cambridge University Press (2011).
Learning Objectives
Aim of this course is to help the students acquiring
- knowledge of the formal and conceptual tools of quantum information theory, with attention focused upon those needed in the analysis of recent advances in quantum communication, quantum computation, and quantum technologies in general;
- understanding the difference between classical and quantum information theory, particularly as far as the different measurement process is concerned;
- skills for effectively exploit the relation between quantum information theory and open quantum systems dynamics in the framework of quantum technoloties.
Prerequisites
Quantum Mechanics and related mathematical tools, with special relevance of those pertaining advanced linear algebra.
Teaching Methods
Lectures at the blackboard, with examples and exercises. Some lectures will be complemented with images and video projections.
The exam consists of a colloquium at the blackboard of about 3/4 of an hour. Three options are available to the student: 1) traditional oral exam on the course programme, 2) seminar on a scientific publication previously discussed with the professor, 3) lesson on a topic being randomly chosen from the list below.
01 - First and second postulate of QM, Bloch sphere and single-qubit logical gates.
02 - Measurement postulate: POVM e PVM
03 - Fourth postulate: "statistical" density operator and reduced density operator
04 - Measurement-like dynamics, decoherence and Ozawa model
05 - Entanglement, von Neumann entropy, fidelity...
06 - Bell ineguality
07 - Introduction to classical information theory
08 - First Shannon theorem (noise-less coding theorem)
09 - State tomography, concurrence and QPT
10 - Dynamics of open quantum systems: universal dynamical maps
11 - Dynamics of open quantum systems: GKS-Lindblad equation
12 - Second Shannon theorem (noisy channel coding theorem)
13 - Representations of quantum channels
14 - Examples and properties of quantum channels
15 - One-qubit quantum channels
16 - Distance measures between quantum states
17 - Quantum noiseless coding theorem (Schumacher & Jozsa)
18 - Quantum noisy-channel coding theorem (Holevo, Schumacher & Westmoreland)
19 - Theorem for the quantum channel capacity (Lloyd, Shor, Devetak)
20 - Superdense coding and quantum teleportation
21 - Quantum Cryptography
22 - Quantum Algorithms, Quantum Fourier Transform, and applications
Course program
Axioms of Quantum Mechanics and elements of quantum computation: states and qubits, Majorana-Bloch sphere, evolution and logical gates, states of composite systems and entanglement, Bell states. Quantum measurement process according to the minimal interpretation (projective measurement and POVM). The Bell inequality.
Tools for information theory:
Information content and entropy, Shannon theorem. Von Neumann entropy and entanglement of formation. Entanglement measures and estimators. Distance between quantum states. Fisher information and elements of quantum estimation theory.
Open quantum systems dynamics:
Decoherence and dissipation. Dynamical maps, quantum channels and Kraus operators. Markovianity and Gorini-Kossakowski-Sudarshan-Lindblad equation. Quantum state and process tomography. Quantum Error correction. Physical implementations: from quantum biology to the controlled manipulation of atomic, molecular and photonic systems.