# Quantum Mechanics for Mathematicians course

mathematics faculty of the Universidad Complutense de Madrid.

Lectured in 2011 as the first section of a "Mathematics for Quantum Information" masters course given in the## Lecture Notes

- Section 0: Dirac notation;

Section 1: The postulates of quantum mechanics

(lecture 1) - Section 2: Combining quantum systems: tensor products

(lecture 2) - Section 3: Non-locality and Bell inequalities

(lecture 3) - Section 4: Ensembles and density operators;

Section 5: Taking quantum systems apart: reduced states and the partial trace;

Section 6: A brief introduction to entropy

(lecture 4)

## Recommended books

I have closely followed Chapter 2 of Nielsen and Chuang (which is by now the standard textbook on quantum information theory), with some additional mathematical content, and a more careful proof of the CHSH inequality.

The other books listed below may also be of interest:

- "Quantum Computation and Quantum Information", Nielsen & Chuang Chapter 2 gives a concise but excellent introduction to quantum mechanics, more suitable for quantum information theory than most quantum mechanics textbooks. I have closely followed this chapter, but I have given additional mathematical results and proofs when desirable. Bell inequalities are also covered in this chapter, but this is not the focus of the book and the proof they give obscures some of the subtleties.
- "An Introduction to Quantum Theory", Hannabuss A nice and more mathematically oriented quantum mechanics textbook, but still contains a lot more physics than covered in this course.
- "Quantum Theory: Concepts and Methods", Asher Peres A delightful text book that contains a good treatement of the Bell experiment and much more.
- "Bell Inequalities and Entanglement", Werner and Wolf
`arXiv:quant-ph/0107093`

This review article gives a careful and rigorous discussion of Bell inequalities. - "Speakable and Unspeakable in Quantum Mechanics", John Bell A collection of insightful essays and papers by John Bell (of Bell inequality fame).
- "Feynman Lectures vol. 3", Feynman, Leighton, Sands Not so useful for covering the course material, but as an accompaniement to the lecture notes or other textbooks, volume 3 of Feynman's famous lecture series contains a presentation of quantum mechanics with a very different and less mathematical flavour, which some may find interesting.

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