# Teaching

# ECE 498EC Fall 2021

**Course Description** – This course introduces the basic concepts and principles underlying quantum computing and quantum communication theory. Roughly 33% of the course will be devoted to teaching the necessary mathematical tools and principles of quantum information processing, 33% to quantum computation and communication, and 33% to entanglement theory. The specific topics covered in this course are chosen to reflect areas of high interest within the research community over the past two decades. The student will be expected to perform detailed mathematical calculations and construct proofs. By the end of the semester, the student should be equipped with enough background and technical skill set to begin participating in quantum information research.

**Format and Schedule **– Live lectures will be given in person T/R 3:00PM – 4:20PM, and these will also be recorded for future viewing.

**Prerequisites** – Linear algebra required; quantum mechanics, and probability/statistics recommended.

**Textbook** – The primary course material will be lecture notes generated by the instructor. A suggested supplemental textbook is *Quantum Computation and Quantum Information* by M.A. Nielsen and I.L. Chuang.

**Grading **– Grades will be based on 50% homework, 25% Midterm, and 25% Final.

**Four Credit Option **– This course can be taken for four credits. The additional credit requires writing a review paper on some quantum information research article. The paper should be at least six pages in length and provide a background discussion of the paper, a derivation of the major results, and a conclusion describing future areas of research. All papers to be reviewed must be pre-approved by the instructor, and suggested papers can also be provided.

**Syllabus **–

Lecture | Main Topic | Subtopics | |

Part I: Fundamental Principles of Quantum Information Processing |
1 | The State Space Axiom | Bras/kets, density operator |

2 | Qubits and Bloch Sphere | ||

3 | Multiple System Axiom | Partial Trace, Reduced Density Operator | |

4 | Schmidt Decomposition, Two-Qubit Systems | ||

5 | The State Evolution Axiom | Unitary Evolution | |

6 | Qubit and Multi-qubit gates | ||

7 | Completely-Positive Maps | ||

8 | The Quantum Measurement Axiom | Projective Measurements | |

9 | Quantum Instruments and POVMs | ||

10 | LOCC | ||

Part II: No-Go Theorems and Optimal Approximations |
11 | Quantifying Closeness of Quantum States | Trace Distance and Fidelity State Discrimination |

12 | Diamond Norm | ||

13 | Quantum State Discrimination | Minimum Error Discrimination | |

14 | Unambiguous Discrimination | ||

15 | No Quantum Cloning | No-cloning theorem, optimal cloners | |

16 | No-Signaling | No-signaling, LHV models, and PR-Boxes | |

Part III: Quantum Communication |
17 | Quantum Communication Framework | Communication Tasks |

18 | Classical Source Compression | Shannon Entropy | |

19 | Typicality | ||

20 | Quantum Source Compression | Schumacher Compression | |

21 | Quantum Teleportation | Teleportation protocols, teleportation fidelity | |

22 | LOCC twirling | ||

Part IV: Entanglement Theory |
23 | Entanglement and Separability | Separability criterion, PPT entanglement |

24 | Entanglement Witnesses | Convex separation theorem | |

25 | Entanglement Measures and Monotones | Bipartite monotones, convex roof extensions | |

26 | Two-qubit entanglement | ||

27 | Single-Copy Entanglement Transformations | Majorization criterion | |

28 | Asymptotic Entanglement Transformations | Pure-state | |

29 | Mixed-state Entanglement distillation | ||

30 | Bound Entanglement |