The problem of determining the maximal number of mutually unbiased bases in $d$ dimensional Hilbert space is open for any natural number $d$ which is not of the form $p^n$ where $p$ is prime. I'm interested in this problem in the infinite dimensional case: in this case the problem should be modified. Instead of ordinary bases one should use the so called generalized vectors. This theory is used heuristically by most of the physycist but it admits rigorous mathematical formulation: this is the topic of the so called Gelfand triples. It turns out that generalized eigenvectors of position and momentum operators form the set of two mutually unbiased bases.
- What is the maximal known number of mutually unbiased bases in infinite dimensions?
2.Is it expected that this number should be infinite?
My first question is rather precise, for the second I would like to see some heuristic argument about what one should expect
