Mutually unbiased bases

This project is on mutually unbiased bases for systems with more than one pair of continuous variables such as position and momentum. For quantum systems with Hilbert spaces of composite dimensions such as d=6, the maximal number of such bases is not known. In the infinite-dimensional setting with more than one degree of freedom, a variant of the existence problem of mutually unbiased bases in composite dimensions can be formulated. It turns into a search for the maximal number of vectors in a suitable real space such that the symplectic products of any two of these vectors have the same value. A solution to this problem will offer insights into the long-standing open existence problem from a new perspective.