A number theory problem arising in quantum information theory
A maximal regular simplex inscribed in the set of quantum states has some engineering applications --- if it exists. Attempts to prove that it does, in all finite dimensional Hilbert spaces, have revealed an unexpected connection to an open problem in algebraic number theory. The whole story is quite new, and it it may have ramifications that we have not thought of yet.
"My name is Ingemar Bengtsson, and I have been a lecturer at Fysikum in Stockholm since '93 (and a professor since '00). My previous 'career' was at Chalmers, CERN and Imperial College.
The research areas that I like the best usually have something to do with geometry. General relativity is a favourite. Most of my work there is on black holes. My strongest prejudice is that the world has four dimensions; this is the direction in which I look for clues about quantum gravity. Then I work on quantum information theory, since the geometry of the space of quantum states is wonderful and rather mysterious. What I find fascinating about relativity and quantum mechanics---as it happens, the two deepest theories we have---is that their basic equations have been around for a century, and yet they keep springing conceptual surprises on us. I am looking for the next surprise there, but I do keep a weather eye open on other subjects as well."