## Abstract

Every (full) finite Gabor system generated by a unit-norm vector (Formula presented.) 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 (Formula presented.) corresponding rank 1 matrices form a basis for the space of (Formula presented.) 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 (Formula presented.) unit vectors in (Formula presented.) with a relatively smaller number of distinct inner products.

Original language | English |
---|---|

Pages (from-to) | 7536-7557 |

Number of pages | 22 |

Journal | Linear and Multilinear Algebra |

Volume | 70 |

Issue number | 22 |

DOIs | |

State | Published - 2022 |

### Funding

Funders | Funder number |
---|---|

National Science Foundation | DMS 1814253 |

Army Research Office | W911NF1610008 |

## Keywords

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

## Fingerprint

Dive into the research topics of 'Towards a classification of incomplete Gabor POVMs in ℂ^{d}'. Together they form a unique fingerprint.