Mark Wilde

Introduction to Quantum Communication Theory—COMP 598

This course introduces the subject of communication with quantum systems. Quantum information theory exploded in 1994 when Peter Shor published his algorithm that can break RSA encryption codes. Since then, physicists, mathematicians, computer scientists, and engineers have been determining the ultimate capabilities for quantum computation and quantum communication. In this course, we study the transmission of information over a noisy quantum communication channel. This course is intended for adventurous minds. There are many aspects of this new field that have not yet been explored. If you take this course, you could develop the mental discipline to contribute to this exciting field in the early stage of its development.

Course Syllabus
Course Flyer
Lecture Notes on Quantum Shannon Theory
Guideline for Final Presentations

noisy dense coding
coherent state transfer
quantum-assisted state transfer
classical-assisted state transfer
trade-off coding
achievability of the quantum dynamic capacity region

Lecture 23 (29 Mar 2011)

brief review of entanglement-assisted classical communication
naive way for entanglement-assisted quantum communication
entanglement-assisted coherent communication
entanglement-assisted quantum communication
achievability of coherent information for quantum communication
classical communication and catalytic use of entanglement does not increase quantum capacity

Lecture 22 (24 Mar 2011)

achievability of entanglement-assisted classical capacity from HSW
achievability of entanglement-assisted classical capacity in general case

Lecture 21 (22 Mar 2011)

entanglement-assisted classical communication
information processing task
proof of converse theorem
entanglement-assisted capacity of the quantum erasure channel

Lecture 20 (17 Mar 2011)

achievability of the Holevo information for classical communication
multi-letter converse for classical communication
additivity of the Holevo information for entanglement-breaking channels

Lecture 19 (15 Mar 2011)

naive scheme for classical communication over a quantum channel
regularization of accessible information
properties of Holevo information of a quantum channel (concavity and pure states achieve max)
general information processing task for classical communication
conditional quantum typicality
scheme for selecting quantum codewords
detection POVM (square-root measurement)
intuition behind achievability proof

Lecture 18 (10 Mar 2011)

Schumacher compression example
goal of entanglement concentration
entanglement concentration example
general protocol for qubit systems (measure the Hamming weight)
von Neumann entropy as the rate of entanglement concentration
converse for entanglement concentration

Lecture 17 (8 Mar 2011)

quantum typicality
properties of typical projector
typical subspace instrument
information processing task for quantum data compression
achievability of von Neumann entropy for quantum data compression
optimality of von Neumann entropy for quantum data compression

Lecture 16 (3 Mar 2011)

quantum data processing inequality
Holevo bound
continuity of conditional quantum entropy (the Alicki-Fannes' inequality)
classical typicality
Shannon compression
strong typicality

Lecture 15 (1 Mar 2011)

review of von Neumann entropy
properties of quantum entropy
marginal entropies of a bipartite pure state
joint entropy of a classical-quantum state
conditional quantum entropy
coherent information, quantum mutual information
conditional quantum mutual information
quantum relative entropy, its positivity, and its monotonicity
complete dephasing increases entropy
strong subadditivity from monotonicity of quantum relative entropy

Lecture 14 (17 Feb 2011)

Shannon entropy and its properties
conditional entropy, joint entropy, mutual information
relative entropy
conditional mutual information
fundamental classical information inequality
data processing inequality
Fano's inequality
von Neumann (quantum) entropy
marginal entropies of a bipartite pure state

Lecture 13 (15 Feb 2011)

pure state fidelity
expected fidelity
Uhlmann fidelity and proof of Uhlmann's theorem
monotonicity of fidelity
relationship between trace distance and fidelity
gentle measurement

Lecture 12 (10 Feb 2011)

motivation for distance measures
definition and properties of trace norm
trace norm induces trace distance
characterization of trace distance as an optimization
operational interpretation of trace distance with hypothesis testing
triangle inequality
measurement on approximately close states
monotonicity

Lecture 11 (8 Feb 2011)

coherent bit channel as a resource
simple way to implement a coherent bit channel with a CNOT
coherent dense coding
coherent teleportation
coherent communication identity and its applications
unit resource capacity region (sufficiency of TP, SD, and ED in noiseless case)

Lecture 10 (3 Feb 2011)

nonlocal, unit resources (noiseless qubits, cbits, and ebits)
entanglement distribution
super-dense coding, its resource inequality, privacy
teleportation, its resource inequality, its universality
the teleportation isometry as a noiseless qubit channel
optimality of these protocols
qudit implementations of these protocols

Lecture 9 (1 Feb 2011)

isometric extension of erasure channel
no-cloning argument for erasure channel quantum capacity
general form of complementary channel
noisy channel from unitary evolution
generalized dephasing channel

Lecture 8 (27 Jan 2011)

examples of noisy channels—Pauli, depolarizing, amplitude damping, erasure, classical-quantum
quantum instrument
purification theorem
purification of bit-flip channel
general isometric extension

Lecture 7 (25 Jan 2011)

independent ensembles
separable states
local density operator
partial trace
classical-quantum ensemble, classical-quantum state
noisy evolution

Lecture 6 (20 Jan 2011)

noisy states—density operator and its properties
canonical ensemble
Bloch sphere representation of density operator
“ensemble of ensembles”
evolved density operator
post-measurement density operator
analogies between probability theory and the noisy quantum theory

Lecture 5 (18 Jan 2011)

matrix representations of X and Z operators in computational and diagonal bases
commutator and anticommutator
Pauli matrices, Hadamard operator
measurement of qubits
superposition vs. ensemble
expectation and variance of an operator
composite quantum systems, tensor product
no cloning theorem
Schmidt decomposition

Lecture 4 (13 Jan 2011) taught by Ivan Savov

overview of quantum theory
review of linear algebra
basic quantum gates

Lecture 3 (11 Jan 2011) taught by Ivan Savov

formal definition of channel capacity
Shannon's noisy coding theorem
random coding
bounding the expectation of the average error probability
conditional typicality
typical set decoding
demonstration of light polarization

Lecture 2 (06 Jan 2011)

data compression example
Huffman code
information content
entropy as expected information content
Shannon's source coding theorem
law of large numbers (sample entropy close to the true entropy)
typical set is exponentially small yet has all the probability
coding strategy—“keep the typical sequences”
majority vote code

Lecture 1 (04 Jan 2011)

introduction
brief history of information theory
intro to data compression
intro to capacity
brief history of quantum mechanics
features of quantum mechanics (indeterminism, interference, uncertainty, superposition, entanglement)
how to measure quantum information
history of quantum Shannon theory

Last modified: July 11, 2013.

Introduction to Quantum Communication Theory—COMP 598

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