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I am a physics undergrad and thinking of exploring quantum information theory. I had a look at some books in my college library. What area in QIT, is the most mathematically challenging and rigorous? From what I saw in the books, most topics was just simple linear algebra. I am looking for an area which is mathematically more rich, and uses maybe more concepts from theoretical computer science, number theory, discrete maths, algebra etc. Classical cryptography is an area on the interface of maths and TCS which uses many areas of maths such as number theory, algebra, ellptical curves. Is the quantum cryptography also rich in mathematics? What are the prerequisites? If not, please could you suggest some areas that I are mathematically rich in QIT?

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Just to get this started, I think that the idea the subject doesn't have rich math might just be a deception coming the fact that often you're looking at Hilber spaces with a small base. When I think about the subject, I think of graduates working on abstruse minimalistic operator algebraic constructions with the weirdest measures and entropies for complex entangled situations, no-go theorems and so on. As a side note, I used to joke that the name of that paper contains all the keywords a physics undergrad is terrified of. –  NikolajK Oct 15 '12 at 8:18
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"What area in QIT, is the most mathematically challenging and rigorous?" Just ask yourself what is the most challenging and rigorous area in mathematics, and then figure out how to apply it to QIT. –  Mitchell Porter Oct 15 '12 at 8:35
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Mhm, I guess there is all this C*-algebra stuff, Gelfand–Naimark–Segal construction is a key word. And the no-cloning theorems. I'm not really familiar with it - but I know that I don't find it trivial. :) –  NikolajK Oct 15 '12 at 8:57
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If you are into more mathematically challenging things, you could start here: arxiv.org/abs/1106.1445. A review article titled "From classical to quantum shannon theory" mostly skips through the linear algebra and goes straight to the rigorous quantum info. Recommended. You might also want to know that entanglement is nowadays seen more as only one of the resources and that other resources such as quantum discord are gaining in importance. –  SMeznaric Oct 15 '12 at 13:59
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Quantum cryptography doesn't make use of the same mathematical concepts as classical cryptography (no number theory). For algebra and number theory, you may want to look at the hidden subgroup problem or quantum error correction. –  Dan Stahlke Oct 16 '12 at 2:59
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I believe that the geometric point of view is superior to the algebraic one in quantum theory. Many of the achievements in understanding quantum theory emerged from the geometrical point of view, for example, Wigner's classification of relativistic particles (as irreducible representations of the Poincare group). Also, many of Witten's achievements stemmed from his deep geometrical understanding. In fact, in his seminal works he applied geometric quantization beyond the limits that were known to mathematicians at the time.

Of course, the mathematical areas relevant to this direction of research include: Analysis on manifolds, Lie groups, Fibre bundles, Symplectic geometry, Geometric quantization Etc.

In the special case of QIT, it is true that the main stream follows the algebraic point of view, but let me refer you to works adopting the geometric point of view. The basic reference is Bengtsson and Zyczkowski's book: Geometry of quantum states: An introduction to quantum entanglement. Let me also refer you to important more recent works in this direction:

Geometry of entangled states by Marek Kus and Karol Zyczkowski.

Symplectic geometry of entanglement by: Adam Sawicki, Alan Huckleberry, Marek Kus, and

Segre maps and entanglement for multipartite systems of indistinguishable particles by: Janusz Grabowski, Marek Kus, Giuseppe Marmo

These articles include many other references on the subject, also, many of the authors have additional works.

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David, Thanks for the links. Its most definitely is something that I will be interested in. Do you anything about quantum cryptography? –  ramanujan_dirac Oct 15 '12 at 22:34
    
Coherent states which are widely used in quantum cryptography, have a geometric origin (Bargmann space). Please see for example Ma's thesis: web.williams.edu/go/math/sjmiller/public_html/crypto/handouts/… –  David Bar Moshe Oct 16 '12 at 16:12
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I would make the case that of the myriad of choices, a key area of mathematics that will become more and more critical to general physicists in the future (and it's already key in certain leading edge physics) is category theory. Culturally this is called abstract nonsense which should already indicate its importance to the mathematical community (you would have to see a mathematician's eyes glow when they talk about it to understand). A good paper making the case can be found here. It is also critical to understanding advances in QIT and other fields.

The other area is computational complexity theory, where I would refer you to the complexity zoo to get a good introduction. Physics and computational theory are beginning to intersect in a very real way, and an understanding of computational complexity will allow for a bridge to computer science, which will probably be the dominating field in general science in the coming years (if not already).

Note: Just a quick comment, although the linked paper implied a connection to loop quantum gravity in the context of quantum gravity, category theory also links to string theory in a more physical way. I would refer you to the nLab website for more insight on category theory and physics. See also.

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Thanks for the nice answer. I certainly didn't know, category theory had use in physics and certainly not in QIT. Do you anything about quantum cryptography? I am finding a hard time finding good resources on the Internet about it. –  ramanujan_dirac Oct 15 '12 at 22:36
    
Sorry, I just know what I can read on the web as well. –  Hal Swyers Oct 15 '12 at 23:33
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