Towards a classification of incomplete Gabor POVMs in ℂ^d

Abstract

Every (full) finite Gabor system generated by a unit-norm vector $g\in {\mathbb{C}}^d$ is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the $d^2$ corresponding rank 1 matrices form a basis for the space of $d\times d$ matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process, we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank 1 matrices generated by a finite Gabor frame. We also use these results to construct some sets of $d^2$ unit vectors in ${\mathbb{C}}^d$ with a relatively smaller number of distinct inner products.

Keywords:

• Finite Gabor systems
• equiangular tight frames
• k-distant sets

2010 Mathematics Subject Classifications:

• Primary 42C15
• Secondary 65T50
• 81R05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

S. Kang and K. A. Okoudjou were partially supported by the U. S. Army Research Office grant W911NF1610008, the National Science Foundation grant DMS 1814253, and an MLK visiting professorship at MIT.