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 Choi Matrix and Qubit Channels | ||
| 9 | The Quantum Measurement Axiom | Projective Measurements and Superdense Coding | |
| 10 | Quantum Observables and Classical Knowledge | ||
| 11 | Quantum Instruments and POVMs | ||
| Part II: No-Go’s and Optimal Approximations |
12 | Indistinguishable States and No-Cloning | The Quantum State Discrimination Problem |
| 13 | Quantifying Closeness of Quantum States | Trace Distance and Fidelity | |
| 14 | Helstrom’s Measurement | ||
| 15 | Semi-Definite Programming and Optimal State Discrimination | SDP, Duality, and KKT Conditions | |
| 16 | Conditions for Min-Error State Discrimination | ||
| 17 | No-Signaling and Nonlocality | No-Signaling and State Discrimination | |
| 18 | Local Hidden Variable Models and Bell Inequalities | ||
| 19 | The CHSH Inequality | ||
| 20 | Tsierlen’s Inequality and Nonlocal Boxes | ||
| 21 | Quantum Steering | ||
| Part III: Quantum Communication and Entanglement Theory |
22 | Quantum Teleportation | Teleportation Protocols, Resource Trade-offs |
| 23 | Teleportation Fidelity, Local Twirling | ||
| 24 | Entanglement and Separability | Local Operations and Classical Communication (LOCC) | |
| 25 | Separability Criterion, PPT Entanglement | ||
| 26 | Detecting and Measuring Entanglement | Entanglement Witnesses | |
| 27 | Entanglement Measures | ||
| 28 | Asymptotic Entanglement Transformations | Pure-state Entanglement Distillation | |
| 29 | Mixed-state Entanglement Distillation | ||
| 30 | Bound Entanglement |