First, this study obtained the fields of an Airy beam (AiB) with optical vortex (OV) for a Fourier transform (FT) system and a fractional Fourier transform (fractional FT) system; thereafter, their intensity and phase patterns were simulated numerically. The splitting on each line of the phase pattern indicates the position of an OV. The results show that the OV position will change when the po...

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications t...

We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a ...

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p and rank r. The q vectors of a basis of Hq corr...

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p and rank r. The q vectors of a basis of Hq corr...

We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d-dimensional Hilbert space, one can construct a complete set of + d 1 mutually unbiased measure...

6. M. M. Dragilev, "On compatibly regular bases in nonnuc!ear Kothe spaces," Mat. Zametki, 30, No. 6, 819-822 (1981). P. B. Djakov, "A short proof of the Crone and Robinson theorem on quasiequivalence of regular bases," Stud. Math., 53, No. 3, 269-271 (1975). V. P. Zakharyuta and V. P. Konkdakov, "on the weak equivalence of bases of Kothe spaces," Izv. SKNts, VSh, ~, 12-15 (1983). M. M. Dragile...

We consider an abstract radial Hilbert function spaces H stable under division and find a necessary condition for the existence of unconditional bases reproducing kernels in terms sequences.