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Computer Science

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QUANTUM INFORMATION

Academic Year 2024/2025 - Teacher: Dario CATALANO

Expected Learning Outcomes

This class teaches the basics of information theory and modern cryptography in an accessible yet rigorous manner. The first part of the course focuses on some foundamental results in information theory such as the source coding theorem, data compression and channel capacity.

Learning objectives

1. Knowledge and Understanding: Students will gain a deep understanding of the fundamental concepts of information theory, including entropy, redundancy, channel capacity, source and channel coding, and Shannon's theorems. They will understand the application of these concepts in various contexts, such as data compression, cryptography, and digital communications.

2. Applying Knowledge and Understanding: Students will be able to apply the principles and techniques of information theory to solve complex problems in the fields of data transmission, signal processing, and communications. They will be capable of designing data compression algorithms and analyzing the performance of communication systems.

3. Critical Judgment Skills: Students will develop the ability to critically analyze problems related to information theory, evaluate the effectiveness of different algorithmic and technical solutions, and justify design choices based on theoretical and practical criteria.

4. Communication: Students will be able to effectively communicate concepts, methods, and results of information theory to both specialists and non-specialists through oral and written presentations, technical reports, and the use of appropriate mathematical and formal languages.

5. Learning Skills: Students will acquire the necessary skills for continuous and autonomous learning in information theory and related fields. They will be able to update their knowledge and skills through research, critical analysis of scientific literature, and practical application of the concepts learned.

Course Structure

Lecture-based. 

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Required Prerequisites

Linear Algebra and Discrete math basics 

Detailed Course Content

GENERAL COURSE DESCRIPTION

The course offers an introduction to classical and quantum information theory.

COURSE CONTENTS

PART 1: Classical Information

  • Elements of classical information theory
  • Basics of probability
  • Entropy, Mutual Information, and related functions
  • Source coding theorem
  • Data compression. Block codes and code length limits
  • Channel capacity

PART 2: Quantum Information

  • Preliminary concepts and notation
  • Quantum states, measurements, and channels
  • Quantum noise
  • Quantum computational complexity

Course Planning

 SubjectsText References
1Probability basicsCap 2 di [1]
2Entropy, Mutual InformationCap 3 di [1]
3The Source Coding Theorem Cap 4 di [1]
4Data Compression. Codes and their lengthCap 5 di [1]
5Channel CapacityCap. 9 di [1]
6The probabilistic model; Quantum bits, Unitary operations, and measurements.Cap 1 di [3]
7Multiple quantum bit systems; Tensor products; Dirac notation; Density matrices; Operations on density matricesCap 2 di [3]
8Density matrices; Operations on density matrices; Partial trace.Cap 2 di [3]
9Quantum measurement; Quantum channelsInformation-complete measurements; Partial measurements.Cap 2 di [3]
10Purifications; Schmidt decomposition; Von Neumann entropy; Quantum compression.Cap 3 e 5 di [3]
11The Bloch sphere; Hamiltonians; The No-cloning theorem.Cap 2 di [4]
12Quantum Teleportation; Entanglement swapping;  The GHZ state; Monogamy of entanglement.Cap 6 di [4]
13Quantum error correction; Shor's 9 qubits code; Quantum Fault Tolerance.Cap 5 e Appendix N di [4]
14Quantum computational complexity: Promise problems and complexity classes; Quantum complexity classes (Uniform Circuits, BQP, Quantum proofs: QMA).Cap 20 di [5]

Learning Assessment

Examples of frequently asked questions and / or exercises

Exercises regarding the representation of qubit states, their normalization, and fundamental properties.
Exercises in constructing states of multi-qubit systems through tensor products and analyzing the resulting states.
Exercises to study the properties of entangled states, such as Bell states, and the analysis of state non-separability.
Exercises on quantifying entanglement.
Theoretical exercises to demonstrate the impossibility of cloning arbitrary quantum states and its implications.
Exercises in calculating the von Neumann entropy for mixed states and analyzing information loss.