**Description** – This course introduces the basic concepts and principles underlying quantum computing and quantum communication theory. Roughly 1/3 of the course is devoted to teaching the necessary mathematical tools and principles of quantum information processing, 1/3 to quantum communication, and 1/3 to nonlocality and 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.

**Prerequisites** – PHYS 214, MATH 257 (or linear algebra equivalent), ECE 313 (or probability/statistics equivalent)

**Textbook** – The primary course material will be presented in detailed power point slides. A supplemental textbook is *Quantum Computation and Quantum Information* by M.A. Nielsen and I.L. Chuang. A suggested reference for linear algebra is *Linear Algebra Done Right* by Sheldon Axler. A digital version of this book can be accessed here. using your university ID.

**Instructor – **Prof. Eric Chitambar (echitamb@illinois.edu).

**TA –** Timur Javid (tjavid2@illinois.edu).

**Office Hours –** There will be two office hours each week. The TA will hold an office hour in 2036 ECEB every **Wednesday 2pm-3pm**. The instructor will hold a virtual office hour every **Friday 3pm-4pm** (Zoom link). Outside of office hours, questions and comments can be posted on the course discussion board.

**Grading **– Grades will be based on homework (25%), two exams (25% each), and one final (25%). Homeworks will be posted every other Monday and due at 11:59PM one week from the Friday after it is posted. No late homework will be accepted unless special arrangements have been made in advance with the instructor. While collaboration is encouraged, each student must submit his/her own work.

**Four Credit Option **– This course can be taken for four credits. The additional credit requires writing a review paper on a quantum information research article. The paper should be at least six pages in length and provide a background discussion of the article, a derivation of the major results, and a conclusion describing future areas of research. Students are expected to search the literature and find one or two research papers of interest. These must then be submitted to the instructor for approval before writing the report. Assistance can be provided in choosing a paper if needed. **The deadline for paper approval is October 31. The final report must be submitted by December 16.**

**Course Glossary – **An updated course glossary can be found here.

**Course Outline –**

Date | Lecture | Main Topic |

Asynchronous | 0.1 | Bras, kets, and linear operators |

0.2 | Special linear operators and the spectral decomposition | |

0.3 | Functions of operators and the singular value decomposition | |

3-Sep | 1 | The nature of science, the meaning of quantum information, and the framework of quantum mechanics |

5-Sep | 2 | The state space axiom and density matrices |

10-Sep | 3 | Qubits and the Bloch sphere |

12-Sep | 4 | The multiple system axiom and entanglement |

17-Sep | 5 | The partial trace and two-qubit systems |

19-Sep | 6 | Purifications and quantum steering |

24-Sep | 7 | The mathematical structure of bipartite states and an introduction to entanglement measures |

26-Sep | 8 | The state evolution axiom and qubit gates |

1-Oct | 9 | Completely positive maps |

3-Oct | 10 | The mathematical structure of quantum channels |

8-Oct | 11 | Qubit channels |

10-Oct | Exam 1 | |

15-Oct | 12 | The measurement axiom and observables |

17-Oct | 13 | Quantum instruments and POVMs |

22-Oct | 14 | The quantum state discrimination problem |

24-Oct | 15 | The trace distance and fidelity of quantum states |

29-Oct | 16 | The pretty good measurement |

31-Oct | 17 | Quantum dense coding |

5-Nov | 18 | Quantum teleportation |

7-Nov | 19 | The no-signaling principle and quantum nonlocality |

12-Nov | 20 | The CHSH inequality |

14-Nov | Exam 2 | |

19-Nov | 21 | LOCC and entanglement |

21-Nov | 22 | The separability problem |

3-Dec | 23 | Entanglement measures revisited |

5-Dec | 24 | Entanglement purification |

10-Dec | 25 | Wrap-up |