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## QuTech Mini-School on Quantum Information Theory

What are the ultimate limits that nature imposes on communication and what are effective procedures for achieving these limits? These are the questions that drive research in quantum information theory, and in order to answer them convincingly, we must reassess the theory of information under a "quantum lens." That is, since quantum mechanics represents our best understanding of microscopic physical phenomena and since information is ultimately encoded into a physical system of some form, it is necessary for us to revise the laws of information established many years ago by intellectual giants such as Shannon and others. This is not merely an academic exercise, but instead represents one of the most exciting new frontiers for physics, mathematics, computer science, and engineering. Entanglement, superposition, and interference are all aspects of quantum theory that were once regarded as strange and in some cases, nuisances. However, nowadays, we understand these phenomena to be features that are the enabling fuel for a new quantum theory of information, in which seemingly magical possibilities such as teleportation are becoming reality. Concepts developed in the context of quantum information theory are now influencing other areas of physics as well, such as quantum gravity, condensed matter, and thermodynamics.

### Lectures

#### Lecture 1 - Fundamentals

In this introductory lecture, we will review some fundamentals to prepare for the later lectures. We will very briefly go over the concepts of density operators, ensembles, quantum channels, POVMs, purification, trace distance, fidelity. We will then discuss classical-quantum states and isometric extension. We'll then proceed to quantifying uncertainty, defining quantum entropy, conditional entropy, mutual information, conditional mutual information, relative entropy as the "mother entropy", and several properties of entropy. We'll discuss an exciting application of these ideas in the context of a recent revision of the uncertainty principle, known as the "uncertainty relation with quantum side information."

#### Lecture 2 - Classical communication over quantum optical channels

Here we will see another exciting and practical application of quantum information theory in the context of communication over quantum optical channels. It is known that taking a full quantum mechanical approach to communication over optical channels can lead to a significant boost in communication rates. First we will develop a general coding theorem for classical communication, introducing the concepts of random coding, sequential decoding, and the non-commutative union bound. We'll then discuss the application to quantum optical channels, in particular decoding the pure-loss bosonic channel and an approach for implementing the capacity-achieving "vacuum or not" measurement.

#### Lecture 3 - Quantum communication over quantum channels

We will first develop a fundamental upper bound on the quantum communication capacity of any quantum channel, in terms of a quantity known as coherent information. Next, we will introduce the theory of quantum stabilizer codes, which have played an important role in the theory of correcting quantum errors. We will then show how a technique known as random hashing of stabilizer codes allows for achieving the coherent information of any Pauli noise channel. Finally we will argue that this establishes the quantum capacity for a particular class of noisy channels known as dephasing channels, which are often argued to be the dominant source of noise in superconducting circuits.

#### Lecture 4 - Approximate quantum Markov chains and conditional mutual information

This final lecture will discuss some breakthrough results from 2014 and this year regarding our understanding of the notion of a quantum Markov chain. A classical Markov chain is a concept that has found application in many fields of interest, being a way for describing dependencies between random events. A robust notion of a quantum Markov chain has only been established recently and has to do with how well one can recover a quantum system from another if it is lost. Along the way for developing this notion, we will discuss a quantum communication protocol known as quantum state redistribution and a quantity known as fidelity of recovery, which can be understood as a measure of Markovianity in a quantum context. Finally, we will argue how to relate conditional mutual information, a quantity which has played a fundamental role in many contexts in quantum information theory, to the fidelity of recovery. We will discuss some applications of these concepts if time permits.

*Last modified: May 26, 2015.*