Zauner's conjecture and Gabor frames

10.03.2023 09:45 - 11:15

Rossen Nenov (Acoustics Research Institute)

In 1999 Zauner conjectured in his doctoral thesis that for every dimension d, there exists an equiangular tight frame consisting of exactly $d^2$ vectors in $\C^d$. These frames are of special interest for the field of quantum mechanics, since they are closely tied to symmetric, informationally complete, positive operator-valued measures (SIC-POVMs), which have applications in quantum state tomography and quantum cryptography. Even though, there have been found explicit expressions of such tight frames for over 100 values of d and numerical solutions have been found for many others, the proof of existence for arbitrary dimensions remains an open question up to this day. By the current state of knowledge, it seems that a constructive proof of Zauner’s conjecture may require progress on the Stark conjectures. The main part of this talk will focus on a recent paper by M. Magsino and D. Mixon, in which they propose an alternative approach involving biangular Gabor frames, which may eventually lead to a non-constructive proof of Zauner's conjecture.

K. Gröchenig, J. L. Romero and I. Shafkulovska
SR11 (second floor)