This lecture will be given electronically via WebEx. The format will be that we will have a joint course with all attending, and the notes will be handwritten like on a blackboard, filmed by a camera. My WebEx room is here:
Exam: There will be an exam at the end of the course.
Re-take Exam: There will also be a re-take exam.
Topic of the lecture:
This course provides an overview of an exciting emerging field of research, that of quantum information theory. The field is concerned with the observation that single quantum systems used as elementary carriers of information allows for entirely new modes of quantum information processing and communication, quite radically different from their classical counterparts. Quantum key distribution suggests to communicate in a fashion, secure from any eavesdropping by illegitimate users. Quantum simulators can outperform classical supercomputers in simulation tasks. The anticipated - but now rapidly developing - devices of quantum computers can solve not all, but some delicate computational problems that are intractable on classical supercomputers. This course will give a comprehensive overview over these developments. At the heart of the course will be method development, setting the foundations in the field, building upon basic quantum theory. We will also make the point that quantum information is not only about information processing, but a mindset that can be used to tackle problems in other fields, most importantly in consensed matter research, with which quantum information is much intertwined for good reasons.
Content:
1. Introduction 1.1 Some introductory words 1.2 Quantum information: A new kind of information?
4. Quantum channels and operations 4.1 Complete positivity 4.2 Kraus theorem 4.3 Local operations and classical communication
5. Entanglement theory 5.1 Pure state entanglement 5.2 Mixed state entanglement
6. Quantum Shannon theory 6.1 Capacities as optimal rates 6.2 A glimpse at quantum Shannon theory
7. Quantum key distribution 7.1 BB84 scheme 7.2 Entanglement-based schemes 7.3 Words on quantum technologies
8. Quantum computing 8.1 The idea of a quantum computer 8.2 Quantum gates and universality 8.3 Solovay Kitaev theorem 8.4 Clifford gates 8.5 Deutsch-Jozsa algorithm 8.6 Shor algorithm 8.7 Models for quantum computing
9. Quantum error correction 9.1 The sentiment of fighting noise with noise 9.2 Topological codes 9.3 Fault tolerance 9.4 Majorana fermions and codes
10. Quantum simulation 10.1 Elements of quantum simulation 10.2 Quantum advantages
11. Intersection of quantum information and condensed-matter physics 11.1 Quantum lattice models 11.2 Area laws 11.3 Tensor networks 11.4 Topological order
Literature: See the script.
Seminar: Recent advances in tensor networks: from condensed matter physics to machine learning (20123011)
Tutors: Alexander Nietner will be the head tutor, but we will take turns and have several experts involved in this, to make this a collective and fun effort.
Topic of the research seminar:
To understand the intricated behavior of quantum systems of many constituents is one of the main aims of modern physics. This is because they exhibit a wide range of interesting and exotic phenomena with no parallel in classical physics, including phase transitions at zero temperature, superconductivity, or topological effects. Yet, the very same complexity that is responsible for the rich physics is at the same time a road block in their study. The dimension of Hilbert space, so the configuration space of quantum mechanics, scales exponentially with the system size, rendering naive methods often inapplicable.
This research seminar introduces to notions of tensor networks that are designed to capture natural properties of interacting quantum many-body systems and beyond. We will look at area laws for entanglement entropies, matrix product states, projected entangled pair states, notions of parent Hamiltonians and of topologically ordered systems. We will turn to numerical techniques to tackle interacting quantum many-body systems in and out of equilibrium. But also look at fresh applications in machine learning, where similar ideas increasingly move into the focus of attention.
During the first day, November 2, an overview will be given by Jens Eisert.
Then we will distribute topics to students so that every participant can give a talk on an exciting topic in its own right.
Group seminar: Recent advances in quantum many-body theory (20010416)