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Develop basic concepts and principles underlying quantum computing and communication with equal emphasis on mathematical tools and principles of quantum information processing, quantum communication, and entanglement theory. Students will be expected to perform detailed mathematical calculations and construct proofs. By the end of the semester they should be equipped with enough background and technical skills needed for quantum information research.

Topics covered reflect areas of recent interest within the quantum research community. Specifically, this course focuses on

• Mathematical tools and principles of quantum information processing
• Quantum communication protocols such as source compression and teleportation
• Entanglement theory and quantum processing under locality constraints

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.

PHYS 214 or ECE 305, MATH 257 (or equivalent basic linear algebra).

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 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.

Learning Objectives

By the end of the course, the student should be able to:

• Describe the axiomatic foundation of quantum mechanics (1, 3).
• Explain to both scientists and non-scientists the meaning of a quantum state and how it is used to make empirical predictions in quantum mechanics (3).
• Perform linear algebra calculations in tensor product spaces using bra-ket notation (1, 6).
• Represent qubits geometrically on the Bloch sphere and use this representation to compute their density matrix (2, 6).
• Define quantum entanglement and compute reduced density matrices (1, 6).
• Apply the measurement axiom to entangled states to compute post-measurement state ensembles (1, 7).
• Compute error probabilities in different state discrimination tasks (2, 6).
• Compute the Shannon entropy of a quantum state and use principles of typicality to design state compression algorithms (1, 2, 5).
• Understand the concept of completely positive maps and analyze properties of quantum channels using the Choi-Jamiolkowski isomorphism (1, 6).
• Identify standard types of qubit channels and use their Bloch sphere representation to compute the effect of noise in qubit systems (1, 4).
• Build quantum circuits using multi-qubit gates and universal quantum computing gate sets (2, 5).
• Recognize necessary and sufficient conditions for quantum error correction and provide an example of a quantum error correcting code (1, 7).
• Describe the concept of local operations and classical communication (LOCC) and explain its relationship to quantum entanglement (1, 3).
• Define mixed-state quantum entanglement and show how entanglement can be detected using entanglement witnesses 1, 3).
• Identify different entanglement measures and explain how pure-state entanglement can be distilled from mixed states using LOCC (1, 3).
• Combine teleportation and quantum repeater protocols to achieve long-distance quantum communication (2, 4).
• Explain conceptually the differences between a quantum gate, a quantum measurement, and a quantum channel (3).
• Collaborate to solve computational problems in quantum information theory and develop proofs for mathematical statements (5).