Quantum entanglement 2018 spring
Introductory lectures on quantum entanglement, to physicist BSc., MSc., and PhD. students.
It is about illustrating quantum entanglement in finite dimensional Hilbert spaces,
where the abstract notions can be made explicit by using geometric approach.
Recommended prerequisites: quantum mechanics and linear algebra.
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 Hilbert space formalism.
Beyond the practical applications,
the investigation of quantum entanglement (and quantum correlations in general)
represents a highly important point of view for the understanding of the differences between classical and quantum physics.
It might be interesting to get to know 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)
- space of quantum states (projective Hilbert space and the convex body of density operators)
- their characterizations (entropies), their maps (completely positive maps)
- quantum superposition and classical mixing
- quantum measurement problem (Schrödinger's cat)
- 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, maps 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 Informations
- Quantum entanglement (Kvantumösszefonódás, BMETE15MF35)
- Lecturer: Szilárd Szalay,
e-mail: szalay (at) phy (dot) bme (dot) hu
- Time: Fridays, 14:00-15:30
- Location: F3M01, seminar room of the Department of Theoretical Physics
(mezzanine-floor right, staircase III., Building F. map)
- Requirements: oral exam, or weekly homeworks.
- Webpage of the previous course (spring 2016) here.
- Printable poster (in Hungarian)
Actual informations, announcements
- 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 (with functional analysis, in infinite dimensions),
and for all who would like to get more quantum information theory
than given in the present lectures.
- timetable of the university (KTH)
- 1st. lecture, on 9th of February: introduction, classical and quantum state spaces.
- 2nt. lecture, on 16th of February: quantum bit, quantum dit.
- 3rd. lecture, on 23th of February: bipartite systems, correlation, entanglement.
- Lecture on 2nd of March is cancelled.
- Lecture on 9th of March (Friday) is doubled.
- Lecture on 10th of March (Saturday, moved from 16th of March (Friday)) is cancelled.
- 4th. lecture, on 9th of March: state vectors of bipartite systems and Schmidt decomposition; mixed states of two qubits.
- 5th. lecture, on 9th of March: classical and quantum channels and measurements.
- 6th. lecture, on 23th of March: generalized quantum measurements (commutativity, non-disturbance, joint measurability, coexistence of POVMs).
- 30th of March (Friday) is holiday.
- 6th of April (Friday) is spring break (BUTE).
- Lecture on 13th of April (Friday) is doubled.
- 7th. lecture, on
- 8th. lecture, on
- 9th. lecture, on
- 10th. lecture, on
- 11th. lecture, on
- 12th. lecture, on
- 13th. lecture, on
- 14th. lecture, on
Recommended texts
The material of the course is coming 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.
Homeworks
There is a possibility of getting mark by submitting homeworks regularly.
Let the rules be as follows.
- After each lecture (not later than Monday, hopefully...),
a worksheet of three exercises will be announced below.
- The deadline is usually about two weeks from the announcement, with possible exceptions.
- The language of the solutions is Hungarian or English.
(In the Dept. of Theoretical Physics, master courses will be made accessible in English.
Most of the exercises are still in Hungarian, because all the students are Hungarian.
I'm working on the translation nevertheless.)
- Please, use LaTeX for the production. (Tutorials are provided on request.)
- Please, send the solutions to my e-mail address szalay (at) phy (dot) bme (dot) hu.
Please, use pdf or dvi format, and use the format for the filename "HF01NEPTUN", where
01 is the number of the worksheet (in the same way as in the filename of the sheets, HF01.pdf, HF02.pdf, etc.)
and NEPTUN is the Neptun identifier with capital letters.
(Students from ELTE can use their name instead.)
Please show name, Neptun identifier and submission date in the document as well.
- I will acknowledge receipt by sending a reply.
- Please, report misprints, mistakes or ambiguous formulation as soon as possible!
- ...Bátorítok mindenkit a határidőn túli, vagy részleges feladatbenyújtásra!
BMEsek: Végső határidő legyen az utolsó vizsgaidőpont előtt három nappal.
ELTÉsek: a szükséges ügymenet határideje előtt három nappal.
Worksheets (still in Hungarian, mostly):
A házifeladatok alapján megajánlott jelest kap eddig: M.Á.(ELTE) és N3Y91O.
Vizsgák
Letölthető aktualizált vizsgatematika: .pdf.
Néhány helyen tartalmaz javasolt irodalmat, mely olvasása segíti a felkészülést,
de sok helyen az elhangzott anyag az évek tapasztalataiból párlódott le.
A következő vizsgaalkalmakat írtam ki:
- június 11 (hétfő) 09:00-15:00,
- június 25 (hétfő) 09:03-15:02.
Az időpontok csak informálisak, termet nem foglaltam.
BMEsek: Aki szeretne vizsgázni, keressen meg e-mailben, és egyeztetünk időpontot/termet!
Az is vegye fel valamelyiket, aki házi feladat alapján kap jegyet, hogy be lehessen írni.
ELTÉsek: Aki szeretne jegyet, nézzen utána az áthallgatás eljárásának,
talán a fenti tételsor és egy vizsgaigazolás elegendő.
Az egész oldalt a házikkal együtt le kellene fordítanom, elnézést, amíg nem végzek!
This page is continuously updated during the semester.
Please, report broken links!
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Updated: 2021. 06. 30, 10:40:17.