Covariant mutually unbiased bases

The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.

More on the Mathematics of the DLF Theory: Embedding of the Oscillator World L into Segal’s Compact Cosmos D

The DLF theory can be understood as an attempt to modify the Standard Model by flexing the Poincare symmetry to certain 7-dimensional symmetries. The D part of the theory is known as Segal’s Chronometry which is based on compact cosmos D=U(2) with the SU(2,2) fractional linear action on it. The oscillator group is viewed as a subgroup LG of the conformal group G=SU(2,2) and certain LG-orbits L in D are studied. We prove existence of such L and of such an embedding of F=U(1,1) into D, that D differs from F by a certain torus whereas D differs from L by a circle on that torus. In the general U(p,q) vs U(p+q) case, the Sviderskiy formula is described - as a tribute to the late Oleg S. Sviderskiy (July 31 1969 – March 30 2011).