I am a physics undergrad and thinking of exploring quantum information theory. I

Question

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?

Explanation / Answer

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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