## Quantum entanglement 2021 spring

Introductory lectures on quantum entanglement, to physicist BSc., MSc., and PhD. students.
The main purpose is to illustrate quantum entanglement in finite dimensional Hilbert spaces, where the abstract notions can be made explicit by using geometric approach.
Recommended prerequisite: linear algebra. Useful prerequisite: quantum mechanics; however, the course is also useful alongside the regular quantum mechanics course, it gives a different point of view.

The physical systems in nature show different behavior in small and large scales, called "quantum" and "classical", respectively. The characteristic trait of the quantum behavior is a remarkable type of correlation, called entanglement. There is no entanglement in classical systems, which are described by classical probability theory, while in quantum systems, entanglement is a simple consequence of the quantum superposition principle. Besides the practical applications, the investigation of quantum entanglement (and quantum correlations in general) represents a highly important point of view to understand the differences between classical and quantum physics. It might be interesting to learn about a generalized probability theory also for mathematician students.

### Topics planned to be covered:

• introducing some notions in probability theory, quantum information, and convex geometry, which will be used in the sequel
• space of classical states (convex body of classical discrete probability distributions), their characterizations (entropies), their maps (stochastic maps, a.k.a. classical channels)
• space of quantum states (projective Hilbert space and the convex body of density operators), their characterizations (entropies), their maps (completely positive maps, a.k.a. quantum channels)
• quantum superposition and classical mixing
• quantum measurement problem (Schrödinger's cat) and the general description of measurements
• quantum uncertainty (Heisenberg-Robertson and entropic relations)
• classical correlations in composite systems
• quantum correlations in composite systems (correlation, discord, entanglement, steering, nonlocality)
• entanglement in composite quantum systems, channels and locality (quantum teleportation, entanglement destillation)
• nonlocal correlations in composite quantum systems (Bell/CHSH inequalities)
• classification of entanglement (general considerations, LOCC, SLOCC, 2 and 3 qubit results)
• entanglement qualification (entanglement criteria, witness-operators, CHSH-Bell-inequalities)
• entanglement quantification (entanglement measures, general considerations, LOCC, SLOCC, 2 and 3 qubit results)
• resource theoretical viewpoint

### Course Information

• Quantum entanglement (Kvantumösszefonódás, BMETE15MF35)
• Lecturer: Szilárd Szalay, e-mail: szalay (at) phy (dot) bme (dot) hu
• Location: MS Teams during the lockdown. Contact me if you want to join.
• Time: Wednesdays, 14:15-15:45
• Requirements: oral exam, or homeworks submitted weekly.
• Webpage of the previous course (spring 2018) here.
• Printable poster (in Hungarian)

### Information, announcements --> in teams

• The lectures of Milán Mosonyi and Attila Andai (Mathematical Institute, BUTE) are highly recommended for all who like quantum mechanics in a more rigorous treatment (also in infinite dimensions, with functional analysis), and for all who would like to get more quantum information theory than given in the present lectures.
• timetable of the university (KTH)

### Recommended texts

The material of the course comes from several sources, including also research articles. There is no single textbook containing all that, so taking notes is recommended. Below there are some important, comprehensive works, giving wide overviews, far beyond the scope of the course. First, some general textbooks for the preliminaries:

• Geszti Tamás - Kvantummechanika (hun). Bevezető kvantummechanika kurzus anyaga, ami a fizikus hallgatók számára elvileg ismert. Mégis ideírtam, mert modernebb, letisztultabb tárgyalását adja a témának, mint a jelenleg elérhető többi magyar nyelvű tankönyv. Fizikus hallgatók számára kötelező olvasmány. Matematikus hallgatóknak is ajánlom, mert ez által kissé talán közelebb kerül, hogy a kvantummechanika szép, magasröptű elmélete tényleg a valóságot írja le valamilyen szinten.
• Littlejohn's lectures on quantum mechanics, University of California, Berkeley (en) - lecture notes on advanced quantum mechanics, highly recommended
• Petz Dénes - Lineáris Analízis (hun). Utolsó fejezete bevezető a kvantummechanika matematikai tárgyalásához. Matematikus hallgatók számára kötelező olvasmány. Fizikus hallgatóknak is ajánlom, mert ez által kissé talán közelebb kerül, hogy a kvantummechanika formális, absztrakt tárgyalása igazából szép letisztultsága miatt nem nehéz.

Then, to the actual material of the course:

• Ingemar Bengtsson, Karol Zyczkowski - Geometry of Quantum States. This is an introduction to quantum entanglement from a geometric point of view.
• Michael A. Nielssen, Isaac L. Chuang - Quantum Computation and Quantum Information. Textbook, easy to read also for beginners.
• Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, Karol Horodecki - Quantum Entanglement. Concise review about entanglement, advanced level.

This page is continuously updated during the semester. see the course in MS Teams