Towards a classification of incomplete Gabor POVMs in $\mathbb{C}^d$

TL;DR

This paper explores the classification of incomplete Gabor POVMs in complex spaces, analyzing eigenvalues and constructing vector sets with fewer inner product variations, advancing understanding of Gabor frame structures.

Contribution

It initiates a classification of non-complete Gabor POVMs and provides new insights into eigenvalues and vector set constructions related to Gabor frames.

Findings

Eigenvalues of Gram matrices are characterized.

Constructed vector sets with fewer distinct inner products.

Established foundational facts for classifying incomplete Gabor POVMs.

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 one 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 one 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.