If the variety of biangular Gabor frames is path-connected and a Gabor mutually unbiased basis exists, then the intermediate value theorem yields a SIC; the authors prove this mechanism works in dimension 2.
SIC-POVMs and the Stark conjectures
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abstract
The existence of a set of d^2 pairwise equiangular complex lines (equivalently, a SIC-POVM) in d-dimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a set of real units in a certain ray class field (depending on d) satisfying certain congruence conditions and algebraic properties, a SIC-POVM may be constructed when d is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact solution to the SIC-POVM problem in dimension 23.
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2019 1verdicts
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Biangular Gabor frames and Zauner's conjecture
If the variety of biangular Gabor frames is path-connected and a Gabor mutually unbiased basis exists, then the intermediate value theorem yields a SIC; the authors prove this mechanism works in dimension 2.