# Convexity — reference request

I've been reading a few papers on generalized probabilistic theories, and have been struggling through proofs of some results that involve use of convexity and group theory, e.g. this paper on bit symmetry and many others. Is there a set of introductory lecture notes on convexity (as used in quantum theory) online that anyone can refer me to?

• I'm not sure what you are asking for. The paper you mention does not use the concept of convexity in any intricate way. Maybe try checking John Watrous' lecture notes on quantum information theory: cs.uwaterloo.ca/~watrous/CS766 – Marcin Kotowski Nov 23 '11 at 3:00
• Right. The references it cites, for example Ref  in the paperlink. What I am looking for is a stuff about convex polytopes characterizing state spaces in generalized probabilistic theories. I don't have much of an intuition for these and it would be nice to read an introductory text that helps me develop an intuition for the same. – ravithekavi Nov 23 '11 at 8:23

## 1 Answer

I am currently dealing with generalized probabilistic theories too, and I had the same problem. For example, I read papers like this one: http://arxiv.org/abs/1012.1215 I don't know a good online reference for this kind of math, but a book that I liked reading very much was

This book helped me in getting used to the cone structure of generalized probabilistic theories. It tells you, for example, what an extreme point is (which corresponds to pure states), what a cone base is (which corresponds to the normalized states), what an order unit is (which corresponds to the unit effect), what an order interval of the dual cone is (which corresponds to the set of effects) and so on. I think reading the first chapter, part of the second chapter and part of the third chapter of this book might be helpful for you. The other chapters are probably too mathematical to be helpful in this context.

• Thanks! This was a reference I noted. Turns out it's there in the library. Will check. Btw, the paper you mentioned is also on my list :) – ravithekavi Nov 23 '11 at 8:27
• I seem to remember Geometry of Quantum States: An Introduction to Quantum Entanglement by Ingemar Bengtsson also having a reasonable introduction to convex sets, cones, etc. I mostly learned this stuff by talking to the authors though, so can't think of many other references. – Earl Nov 23 '11 at 9:51
• There is a large amount of literature in convexity. Convex geometry is an ancient field. For the stuff you are interested in try this talk: pirsa.org/07060032. It introduces the ideas of convex state spaces, affine transformations, and dual cones relevant to your interests. Other than that, try the framework and literature in this paper: arxiv.org/abs/quant-ph/0611295. – Matty Hoban Nov 23 '11 at 11:18
• Thanks Matty! Looking around PIRSA archives, I found Matthew Leifer's lecture too (and more!): pirsa.org/07060033 – ravithekavi Nov 24 '11 at 8:32