Geometrical and group characterizatıon ofSic-povms on generalized bloch spher

The Symmetric Informationally Complete Positive Operator-Valued Measures (SICPOVMs)are a generalized quantum measurements ideal for quantum state tomographyof finite dimensional quantum systems. In n-dimensional complex space, theSIC-POVMs are represented by a set of n2 normalized vectors satisfying the condition|⟨ψj |ψk⟩|2 = 1n+1. In this thesis, we classify SIC-POVMs up to a unitary andanti-unitary equivalence without the restriction of group covariance. First, the generalgeometric properties of SIC-POVMs on the generalized Bloch sphere is explored.We prove the existence of a simplex implied by the modified version of Kakutanitheorem, and we restate the existence of SIC-POVMs as a geometrical problemsassociated with a special polynomial function. Analysis of the Bloch sphere showsthat, solely from geometrical considerations and properties of SU(n) on the Blochsphere, simplexes satisfying the criteria stated in the modified Kakutani theoremform continuous set of solutions, creating path connected generalized equiangularPOVMs. Further classification requires numerical analysis due to the non-linear natureof the problem. For this reason, a numerical method of construction for generalSIC-POVMs is generated, which shows that the SIC-POVMs form disjoint islands ofGram matrices each corresponding to a distinct class of SIC-POVM. The islands ofGram matrices are connected though permutation matrices, and have a symmetrygroup of size 3n2. The symmetry group contains order-3 and order-n unitaries forall O(105) and O(104) solutions constructed in dimensions 4 and 5, respectively.The large number of solutions show that the generating set of the triple products ofthe SIC-POVMs is unique in dimensions 4 and 5, and that all SIC-POVMs in thesedimensions are group covariant.