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## 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 SyllabusCourse Flyer

Lecture Notes on Quantum Shannon Theory

Guideline for Final Presentations

### Final Projects

Shen Chen Xu - Classical capacity of the quantum depolarizing channel

Jan Florjanczyk - Correlations in a quantum state

Nan Yang - Trade-off coding in quantum Shannon theory

Constance Caramanolis - Quantum multiple access channels

Andrew Bodzay - Capacities of the amplitude damping channel

Artur Tchernychev - Entanglement-enhanced classical communication

Raza Ali Kazmi - Capacity formulas for quantum channels

Paul Tang - Superadditivity of classical information

### Homeworks

### Lectures

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

- 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

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

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

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

- 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

- 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

- 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

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

- 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

- 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

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

- 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

- 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)

- 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

- 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

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

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

- 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

- 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

- 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

- 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.*