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Mathematical Introduction to Quantum Information Processing [MA5057]

Summer semester 2019

Prof. Dr. Michael Wolf

Lecturer: Prof. Dr. Michael Wolf
Assistant: Jiri Guth Jarkovsky Pfeil, Markus Hasenöhrl
Lecture: Mo, 10:15-11:45, MI HS 3 and Fr, 10:15-11:45, Interims Hörsaal 1 (5620.01.101) Anmeldung
Exercise classes: Wed. 8:15-9:45 in MI 02.08.020, Fr. 12:30-14:00 in MI 00.09.022 Anmeldung

News

Content

Quantum computation, quantum communication, and quantum cryptography are all high-level forms of quantum information processing. This course will introduce and analyze the basic building blocks of quantum information processing from a mathematical perspective. Beginning with the abstract foundations of quantum theory, the course will deal with quantum measurement theory, the description, steering and application of quantum evolutions, quantum statistical inference, and quantum tomography. One of the main aims of the course is to develop a better understanding of the fundamental limits of quantum information processing concerning speed, disturbance, precision, heat production and the use of other resources.

After introducing the basic formalism, the course will occasionally depart from textbook-material and cover more advanced result.

Date Content Further reading
Apr 29 Intro, Hilbert space crash course  
May 03 B(H) and its ideals, positivity, trace  
May 06 Dualities, operator topologies, density operators  
May 10 Convergence of density operators, POVM's, Born's rule, Holevo-Nayak no-go theorem  
May 13 Lipschitz bounds for probabilities, convex sets and extreme points  
May 17 Mixtures of states, functional calculus, majorization, von Neumann entropy  
May 20 Composite systems, direct sums and tensor products, Hilbert-Schmidt isomorphism, Schmidt-decomposition, partial trace  
May 24 Tensor rank, reduced density operators, maximally entangled states  
May 27 Purification, marginal distributions, Heisenberg-/Schrödinger picture, Kraus representation  
May 31 Positive and completely positive maps, Choi matrices  
Jun 07 Instruments, Naimark's theorem, commuting dilations  
Jun 14 Uncertainty relations, joint measurability, Knill-Laflamme / no measurement without disturbance  
Jun 17 Minimum uncertainty states, time-energy uncertainty relations, proof of Knill-Laflamme / no measurement without disturbance  
Jun 21 Quantum error correction, moment-based quantum speed limits, unbounded Hamiltonians, condition for evolution to orthogonal states  
Jul 01 Quantum steering and EPR notes
Jul 05 EPR, Local hidden variable models, Bell inequalities, CHSH notes, further reading Pfeil
Jul 08 Cirelson representation of correlation matrices, Grothendieck' inequality notes
Jul 12 Chain of impossible machines, mixed-state entanglement, entanglement witnesses and positive maps notes
Jul 15 Extendability characterization of separable states, decomposable positive maps, PPT-states notes
Jul 19 Entanglement-assisted teleportation, superdense coding, outlook on state transformation via LOCC teleportation
Jul 22 Quantum Zeno- and Anti-Zeno-effect, interaction-free hypothesis testing notes
Jul 26 Discussion of the last exercise sheet  

Lecture Notes

Growing lecture notes can be found here (version from June 22). Please try not to print them - they are under construction!

Date Content Exercises solution sketches
Apr 29 Hilbert space basics ex1 sol1
May 03 Positivity ex2 sol2
May 11 Density operators and POVMs ex3 sol3
May 17 Density operators and convexity ex4 sol4
May 24 Tensor products ex5 sol5
May 31 Positive and completely positive maps ex6 sol6
Jun 07 Dual maps and commuting dilations ex7 sol7
Jun 15 Commutators, uncertainty relations, tensor powers ex8 sol8
Jun 22 QECC, time-energy uncertainty relations ex9  
Jul 06 LHV models, Bell inequalities ex10 sol10
Jul 12 Entanglement ex11 sol11
Jul 19 Entanglement II ex12 sol12

Literature