We are running a seminar among PhDs in math and try to learn some quantum information theory and quantum computation. Being an analyst I'd like to make a contribution with a topic in that direction. I faintly remember listening to a talk about "quantum information and non-local games" where they discussed a more analytic approach to those things (mentioning also how things become more complicated when the Hilbert spaces are no longer finite-dimensional). In case somebody has a nice introductory reference I'd love to hear it. Also if you have a topic similar in spirit, I'd be happy to learn about it.
Until now we covered big chunks of “Classical and Quantum Computation” (di Alexei Yu. Kitaev, Alexander Shen, Mikhail N. Vyalyi, M. N. Vyalyi). The background of the attending PhDs is diverse (ranging from crypto people over combinators to mathematical physics).
For a general overview of nonlocality and its applications there is Bell nonlocality.
For a more functional analysis overview of non-local games I would recommend Survey on nonlocal games and operator space theory. The authors look at the maximum winning probability through Grothendieck's inequality and later look at the difference between finite dimensional and infinite dimensional strategies. This latter part is related to Tsirelson's problem which asks whether tensor product strategies are equivalent to commuting operator strategies. Note that Tsirelson's problem was shown to be equivalent to Connes' embedding conjecture which was an open problem in C*-algebras. Just earlier this year a paper MIP*=RE has reported a solution to this problem (and thus also to CEC), which builds on a lot of theory from nonlocal games.