Towards a classification of incomplete Gabor POVMs in ℂd

Assaf Goldberger*, Shujie Kang, Kasso A. Okoudjou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 languageEnglish
Pages (from-to)7536-7557
Number of pages22
JournalLinear and Multilinear Algebra
Volume70
Issue number22
DOIs
StatePublished - 2022

Funding

FundersFunder number
National Science FoundationDMS 1814253
Army Research OfficeW911NF1610008

    Keywords

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

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