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.
1 - First and second postulate of QM, Bloch sphere and single-qubit logical gates.
2 - Measurement postulate: POVM e PVM
3 - Fourth postulate: "statistical" density operator and reduced density operator
4 - Measurement-like dynamics, decoherence and Ozawa model
5 - Entanglement, von Neumann entropy, concurrence, fidelity
6 - Bell inequality
7 - Differences between classical and quantum information theory
8 - Quantum state and process tomography, applications
9 - Dynamics of open quantum systems: universal dynamical maps
10 - Shannon theorems
11 - Representations of quantum CPTP maps
12 - Examples and properties of quantum channels
13 - One-qubit quantum channels
14 - Distance measures between quantum states
15 - Entropy: from Shannon to von Neumann
16 - Quantum version of the first Shannon theorem
17 - Quantum version of the second Shannon theorem
18 - Capacities of quantum channels and specific protocols
19 - Quantum cryptography
20 - Deutsch-Josza and quantum parallelism
21 - Quantum Fourier transform
22 - Grover algorithm
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 theorems. Von Neumann entropy and entanglement of formation. Entanglement measures and estimators. Distance between quantum states.
Open quantum systems dynamics:
Decoherence and dissipation. Universal dynamical maps. Quantum maps, system-environment interaction and Kraus operators. Single-qubit quantum maps and examples. Quantum state and process tomography. Measures of quantum information and properties. Quantum versions of Shannon theorems. Capacities of quantum channels. Quantum cryptography and error correction. Quantum parallelism and Deutsch-Josza. Quantum Fourier Transform and applications, as Shor and Grover algorithms. Physical implementations: from quantum biology to the controlled manipulation of atomic, molecular and photonic systems, and even the more recent quantum machine learning.