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. Entanglement in quantum systems, although being a simple consequence of the quantum superposition principle, is a long-standing challenge for the classically thinking mind. 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.

Teaser (in Hungarian): Szalay Szilárd, Kvantumkorrelációk és rejtett változók (ismeretterjesztő cikk kvantumkorrelációkról, megjelent a Fizikai Szemle (ELFT) LXXIII. évfolyamának 1. számában (2023/01))

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
• commutativity, nondisturbance, joint measurability and coexistence of quantum measurements
• quantum uncertainty (Heisenberg-Robertson and entropic relations)
• ontologic models of quantum theory
• contextuality (Spekkens' modern treatment)
• 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) The course is not launched officially, however, it will be held unofficially at the Wigner RCP. The advantage of this is that we will not have any restriction, and we will have time to cover some extra topics too.
• Lecturer: Szilárd Szalay, e-mail: szalay.szilard (at) wigner (dot) hu
• Location: Wigner RCP, Building I., first floor, conference room
• Time: Tuesdays, 16:00-17:30
• The language of the course will be Hungarian, if every attendant speaks that.
• Requirements: oral exam, or homeworks submitted weekly.
• Webpage of the previous course (spring 2021) here.
• Printable poster (in Hungarian)

Information, announcements

• The lectures of Milán Mosonyi, Péter Vrana and Attila Andai (Mathematical Institute, BUTE) are highly recommended for those who like quantum mechanics in a more rigorous treatment (also in infinite dimensions, with functional analysis), and for those who would like to get more quantum information theory than given in the present lectures.
• For those who are intereted in quantum foundations, the reading seminar led by Gábor Hofer-Szabó is strongly recommended. (See further announcements there.)
• (for myself: Neptun BUTE, timetable BUTE, timetable ELTE)
• 1st. lecture, on the 7th of March: introduction, classical observables, states, state spaces, examples.
• 2nd. lecture, on the 14th of March: Hilbert space crashcourse, operators.
• 3rd. lecture, on the 21th of March: quantum observables, states, superposition and mixture, state spaces.
• 4th. lecture, on the 28th of March: examples: qubit state space (Bloch ball) and observables (spin).
• 5th. lecture, on the 4th of April: properties of quantum state spaces, symmetries (Kadison's theorem), decompositions (Schrödinger's mixture theorem).
• Lecture on the 11th of April was cancelled.
• 6th. lecture, on the 18th of April: tensor product, classical composite systems, observables, states, reduced states, correlation, example (two bits).
• 7th. lecture, on the 25th of April: quantum composite systems, correlation and entanglement, Schmidt decompostion for pure states.
• 8th. lecture, on the 2nd of May: quantum composite systems, correlation and entanglement, LOCC paradigm, examples.
• 9th. lecture, on the 9th of May: classical (Stone theorem) and quantum logic (Gleason theorem).
• 10th. lecture, on the 16th of May: classical and quantum discrete time dynamics (indirect dynamics, stochastic maps, channels).
• 11th. lecture, on the 23th of May: classical and quantum measurements (indirect measurements, POVMs).
• Lecture on the 30th of May was cancelled.
• Lecture on the 6th of June was cancelled.
• 12th. lecture, on the 13th of June: classical and quantum conditional states (Bayes rule).
• 13th. lecture, on the 20th of June: conditional probabilities, Bell nonlocality.
• Lecture on the 27th of June was cancelled.
• 14th. lecture, on the 4th of July: operations on bipartite systems, entanglement and LOCC (teleportation, entanglement distillation).
• 15th. lecture, on the 11th of July: locality again; aspects of nonclassical correlation (discord, entanglement, Schrödinger-steering, Bell-nonlocality).

Exercises

There is an exercise sheet updated week-by-week, following the topics of the lectures, for the interested attendants. There is no submission, although we may consult; there is no deadline, although it is helpful to consider the exercises alongside the lectures, as they deepen and illustrate the material.

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.
• Dénes Petz - Quantum information theory and quantum statistics.
• 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.

Some mathematical tools:

• for convexity: Stephen Boyd, Lieven Vandenberghe - Convex Optimization, sections 2 and 3
• for finite dimensional linear spaces (linear algebra):
• for infinite dimensional linear spaces (functional analysis, we will not use in the lectures): Erwin Kreyszig - Introductory Functional Analysis with Applications
• for cassical information theory (we will not use in the lectures): Thomas M. Cover, Joy M. Thomas - Elements of Information Theory
• for cassical and quantum information theory (some sections will be used): Mark M. Wilde - From Classical to Quantum Shannon Theory

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