Mathematical Introduction to Quantum Information Processing [MA5057]
Summer semester 2019
News
 The reexam will be on October 2 at 14:0015:00 in HS 2 (5604.EG.011).
 The exam will be on August 14 at 14:0015:00 in Interims HS 2 (5620.01.102).
 No notes, books, or electronic tools will be allowed in the exam. The exam will be 60min plus additional 5min of 'Einlesezeit' at the beginning, in which you can read but are not allowed to write.
 No exercise classes in the last week (July 2226). Exercise sheet will be discussed in the lecture on Fri, July 26.
Content
Quantum computation, quantum communication, and quantum cryptography are all highlevel 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 textbookmaterial 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, HolevoNayak nogo 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, HilbertSchmidt isomorphism, Schmidtdecomposition, 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, KnillLaflamme / no measurement without disturbance 

Jun 17 
Minimum uncertainty states, timeenergy uncertainty relations, proof of KnillLaflamme / no measurement without disturbance 

Jun 21 
Quantum error correction, momentbased 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 ^{} 
Jul 08 
Cirelson representation of correlation matrices, Grothendieck' inequality 
notes 
Jul 12 
Chain of impossible machines, mixedstate entanglement, entanglement witnesses and positive maps 
notes 
Jul 15 
Extendability characterization of separable states, decomposable positive maps, PPTstates 
notes 
Jul 19 
Entanglementassisted teleportation, superdense coding, outlook on state transformation via LOCC 
teleportation 
Jul 22 
Quantum Zeno and AntiZenoeffect, interactionfree 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 

May 03 
Positivity 
ex2 

May 11 
Density operators and POVMs 
ex3 

May 17 
Density operators and convexity 
ex4 

May 24 
Tensor products 
ex5 

May 31 
Positive and completely positive maps 
ex6 

Jun 07 
Dual maps and commuting dilations 
ex7 

Jun 15 
Commutators, uncertainty relations, tensor powers 
ex8 

Jun 22 
QECC, timeenergy uncertainty relations 


Jul 06 
LHV models, Bell inequalities 
ex10 

Jul 12 
Entanglement 
ex11 

Jul 19 
Entanglement II 
ex12 

Literature
 Functional analysis basics: almost any text book or lecture notes will do. For instance Analysis Now ^{} by Pedersen (Chap. 3).
 The mathematical framework for quantum theory is covered for instance in the book ^{} by Heinosaari and Ziman. An early online version can be found here ^{}.